How can there be so many differences between schools of Buddhism? How can we tell which one is the most accurate to avoid having a wrong view?

At times when some of the

At times when some of the schools were possibly passing on doctrines that were perhaps, in some aspects, lacking, misleading, in excess, misunderstood, or possibly erroneous, that would be a critical moment in history when a major effort is called for in order to remedy the situation through debate, council, doctrinal adjustment, purification, reinterpretation, or back-to-the-basic movement--despite our human limitation. As we learn from other courses, the subsequent Mahayana movement and Nagarjuna's unique role in the history may also be explained in this context. Nagarjuna, who pointed out corrected the errors of some other schools, has simultaneously been much credited with the rise of Mahayana/Vajrayana schools, and is considered to be one of the patriarch of many schools that appeared afterwards, and singularly regarded as the "Second Buddha", without whom there might have been no Mahayana/Vajrayana schools as we know them today.

"...Nagarjuna is here referring to two extremes, i.e., permanence (nitya) and impermanence (anitya), this latter being the momentary destruction (ksana-bhanga) advocated by the Buddhist metaphysicians. The former represents the Sarvastivada point of view; the latter, the Sautrantika." (p. 236)
"Nagarjuna's attention is now directed at the Sautrantika view, for it is that which finally contributed to the 'personalist theory' (pudgala-vada) of the Vatsiputrriyas." (p. 299)
"Sautrantika, denying self-nature in phenomena, surreptitiously introduced a conception of self or person (attman, pudgala) in a human personality."
(p. 23, Mūlamadhyamakakārikā of Nāgārjuna, by Kalupahana)

When taken chronologically, the nature of the differences that occurred among the pre-Mahayana schools might not resemble--or be significantly different from--those that appeared during the Mahayana emergence. In connecting the nature of the Mahayana diversity vs. pre-Mahayana diversity with the nature of the the Vinaya differences vs. the doctrinal differences, the following remarks might be appreciated:

Some scholars observe that: "'... In Buddhist tradition, 'splitting of the Sangha' always refers to matters of monastic discipline’ (Bechert 1982a: 65)). Sects might as a matter of fact differ on doctrinal matters, and of course doctrinal differences might arise after schism has occurred, which could then differentiate further the groups thus formed. Nevertheless, differences of doctrine would seem to be a matter of the individual group attracted or convinced by them, rather than a monastic sect as such. In theory a monastery could happily contain monks holding quite different doctrines so long as they behaved in the same way.... Although there are a number of different Vinayas that were formulated in ancient India, the differences, while important to the monks concerned, are nevertheless relatively insignificant. Moreover there was no Mahayana Vinaya as such produced in India. Indian Mahayana Buddhist monks and nuns all adhered to Vinaya rules which were formulated by the sects that originated during the early centuries of Buddhism in India."
(p.4, Mahayana Buddhism: The Doctrinal Foundations, by Paul Williams)

On the matter of differences in Vinaya rules, "...they have absolutely nothing to do with issues of Mahayana versus non-Mahayana. There is no such thing as a Mahayana Vinaya. Thus it is extremely unlikely that Mahayana originated as such in any sort of schism.... Once we understand that Mahayana identity is not a matter of the Vinaya and therefore not a matter of publicly significant behaviour in a monastic context, then it becomes perfectly understandable that visitors to India would have seen Mahayana and non-Mahayana monks in the same monasteries. ... If that still seems strange, then one has still not appreciated the inappropriateness of the schism-model, or that supplied by Christian parallels.... I have referred to Mahayana as a vision, a vision of what Buddhism is finally all about, rather than a sect, a school or the result of schism."
(p. 100-101, Buddhist Thought: A Complete Introduction to the Indian Tradition, by Paul Williams)

As we all know, Mahayana as a movement already includes immensely diverse doctrines, and some even consider Vajrayana to be yet another subsequent movement on its own and classify it separately. Here, the sunyata doctrine of Nagarjuna--the "second Buddha" (小釈迦 in Japan, 第二代釋迦 in China)--may be central in understanding the nature of Mahayana/Vajrayana plurality--also in understanding the possible unity in its diversity.

In regard to the question of diversity, those who belong to the Northern traditions also take Mahayana/Vajrayana sources (Sutras) very seriously, and may take their epistemological sources to be comparable to, in some sense, "revelation", direct or "mystical" experience, or divine inspiration, and their origins to be Bodhisattva (e.g. Nagarjuna, Maitreya), Sambhogakaya, Dharmakaya (e.g. Vairocana Buddha), or even Nirmanakaya (Gautama Buddha) handed down to us through other routes--they are not being taken as a sort of man-made fabrications or artistic innovations of human creativity as it were belonging to a literary genre.

The followings might be of interest to the issue at hand; we may take them as we will...

There is "...a justification for the Mahayana sutras internal to the Mahayana tradition itself and separate but not necessarily contradictory to the theory of revelation."
(p. 41, Mahayana Buddhism: The Doctrinal Foundations, by Paul Williams)

The following article may be revealing in understanding the difficulty of translation and correct interpretation when taken in foreign cultures:

"Just as the Chinese had Taoism as their Dharma gate — the home-grown tradition providing concepts that helped them understand the Dharma — we in the West have Romanticism as ours. The Chinese experience with Dharma gates, though, contains an important lesson that is often overlooked. After three centuries of interest in Buddhist teachings, they began to realize that Buddhism and Taoism were asking different questions. As they rooted out these differences, they started using Buddhist ideas to question their Taoist presuppositions. This was how Buddhism, instead of turning into a drop in the Taoist sea, was able to inject something genuinely new into Chinese culture. The question here in the West is whether we will learn from the Chinese example and start using Buddhist ideas to question our Dharma gate, to see exactly how far the similarities between the gate and the actual Dharma go. If we don't, we run the danger of mistaking the gate for the Dharma itself, and of never going through it to the other side."
The Roots of Buddhist Romanticism by Thanissaro Bhikkhu

In spite of thereby admitting, in exchange, its own new shortcomings with new openings for misinterpretation and peculiar temptation for laxity, Mahayana movement as a whole had also gone through a long history of strict epistemological and hermeneutical discernment:

"Buddhists were well aware of the possibility of works being misrepresented as scripture and were concerned not to be led astray from the correct teaching. Thus, no doubt, many contenders for community approval were anathematized as not the word of the Buddha. Others, however, were found suitable and ... certain well-elaborated literary statements were accepted as valid articulations of the perspective of the community. By this process was gradually created a respected body of knowledge. It is essential to understand that this process is by no means limited to mystical religious traditions such as Tantric Buddhism and Saivism, but was a fundamental feature of Indian cultural discourse in virtually all fields. During the same first-millennium period that witnessed the revelation of the Mahayana and Mantrayana scriptures, similarly consensual understandings deriving from years of social experience across the cultural spectrum were crystallized into major statements of theoretical and practical knowledge. Yet, due to the aforementioned axiom that all that is good and true must have been known of old by divine or semidivine beings, these knowledge systems too were attributed to miraculous revelations."
(p. 98 Making Sense of Tantric Buddhism, by Christian K Wedemeyer)

Simsapa Sutta:
"The leaves in the hand of the Blessed One are few in number, lord. Those overhead in the simsapa forest are more numerous. In the same way, monks, those things that I have known with direct knowledge but have not taught are far more numerous [than what I have taught]."

"The two major schools of Buddhism, Theravada and the Mahayana, are to be understood as different expressions of the same teaching of the historical Buddha. Because, in fact, they agree upon and practice the core teachings of the Buddha’s Dharma."
(The Buddhist Schools: Theravada and Mahayana, BuddhaNet)

The diversity as forms of skillful means:
"The foregoing paradigm of ‘skilful means’ pioneered by the Buddha in the early suttas provided a template for all future developments in Buddhist pedagogy. In later Mahayana Buddhism, in particular, ‘skilful means’ (Sanskrit: upaya-kaushalya) – along with wisdom (prajna) – was the defining quality of the Bodhisattva... ‘Skilful means’ also proved pivotal to Buddhism’s successful expansion from India into other Asian countries"
(The Origins of ‘Skilful Means’ in Early Buddhism, by Peter Nelson, BuddhaNet)

"... it was possible for individual Buddhists, at varying speeds, to come to an understanding of the
historical diversity within the tradition in terms of skillful means. Exactly the same applies with respect to the geographical diversification which Buddhism accepted from the start. The individuals influenced by Buddhism in any one context of cultural syncretism come to recognise, at varying speeds, the inward meaning or intention of the expressions which Buddhism adopted. ....The teaching is never dismantled just because one individual has understood its character as skilful means. "
(p. 160, Skillful Means: A Concept in Mahayana Buddhism, Michael Pye)

Two Truths Theory (the diversity of rafts as forms of Conventional Truths)
"Mādhyamaka school, interpreted as holding that any specific thing we could say about the world was at best a “Conventional Truth”, which served as a kind of “raft” to get beyond it to the 'Ultimate Truth'"
"conventional truths: they are to serve as “rafts”, to be clung to temporarily, but only because they are an effective means"
"all possible statements, viewpoints, ideas, concepts, positions are conventional truths"
(Tiantai Buddhism, Stanford Encyclopedia of Philosophy)

Also, since we are examining the problem of diversity under the academic study of Buddhism, I figured it might turn out to be fruitful to apply, to a lesser extent, some of the analysis and theories taken from the topic Problem of Religious Pluralism frequently raised under the branch Philosophy of Religion. Obviously, the "problem" that occurs within the Buddhism tradition is certainly not to the degree seen among the world religions, some ideas or approaches in this field might turn out to be useful even when applying specifically to the Buddhism schools.

Thank you for an intriguing

Thank you for an intriguing question which all of us probably encounter in one way or another. As to the first question, my take is that there were many reasons to the diversity in Buddhism, and I imagine there could be many approaches in examining the question of how the diversity came about. The following is just my 2 cents... take it as you will and please share us your thoughts and insights too.

The differences may be due to the doctrinal (Sutta, Abhidhamma, Sastra, etc.) differences or those in practice (Vinaya, etc.). Obviously the issues of transmission error, mistranslation, and misinterpretation may be raised here. The differences may also be due, in another respect, to geographical separation--as in Southern (Theravada) schools vs. Northern (Mahayana/Vajrayana) schools--or to national boundaries--political and sociological aspects of Buddhism, such as the possible ties with lay communities and state patrons. The additional issues of cultural application/adaptation might be expected to naturally arise. The subject of the course sheds light on how the Abhidharmic diversity came about in the pre-Mahayana period, but it may also shed light on the backdrop behind the Mahayana emergence.

If we chronologically follow from the very beginning, it is recalled that it only took about a hundred years before the first schism occurred, when the Second Council resulted in the first split between Mahasamghikas and Sthaviras. The Theravada account of the schism is extant, and its cause appears to be due to the Vinaya, rather than the doctrinal, differences. As we know, other subsequent schisms seem to have added the doctrinal (Abhidhamma/Abhidharma) differences. When the inquiry is pressed further as to why the first schism had occurred, it might lead us to speculations--and there may obviously be more than a single cause (as we recall Visuddhimagga: "Nor from a single cause arise One fruit or many, nor one fruit from many...")

If we focus on the factor of language (Pali, Sanskrit, Chinese, etc), then the issues of transmission, translation, and interpretation (exegesis, hermeneutics) might be considered. Multiple languages/dialects are appeared to have been used already during the time when Buddha was spreading the Dhamma in India, as the Buddha's native language might have been Magadhi Prakrit but he might have also used other languages/dialects. Evidently, this seems to be what the Buddha had exactly intended to do in considering its positive aspects, as the Pali record show:

If bound to single language, "...the Buddha’s words would not have easily spread outside of India. Thus, to avoid these limitations, the Buddha forbade that his teachings be turned into a Veda, and instead allowed his followers to memorize the Dhamma each in his own language."
(The Buddhist Monastic Code II, the Khandhaka Rules, Translated & Explained by Thanissaro Bhikkhu)

As to the issue of transmission--possibly with other issues,--it appears to have already occurred during the First Council when Purana said, "Gentlemen, the Doctrine and Discipline have been beautifully chanted in chorus by the Elders; but, all the same, I shall maintain what I heard and received from the mouth of the Blessed One exactly as I heard it" (p. 256). There seems to be "from the moment of Buddha's death, there were at least two recensions of the Canon maintained by parties of equal strength."
(p. 257, The First Buddhist Council by D. T. Suzuki)

As to the issue of the correct interpretation, it was already during the time of the Buddha that there were disciples who had misinterpreted his teachings--possibly had spread misleading or erroneous doctrines in consequence. For example, the Pali records show how the Buddha admonishes his disciples who misunderstood the doctrine:

"...the Blessed One said to him, "Is it true, Sāti, that this pernicious view has arisen in you — 'As I understand the Dhamma taught by the Blessed One, it is just this consciousness that runs and wanders on, not another'? ...."And to whom, worthless man, do you understand me to have taught the Dhamma like that?"
(Mahatanhasankhaya Sutta www.accesstoinsight.org/tipitaka/mn/mn.038.than.html)

"Worthless man, from whom have you understood that Dhamma taught by me in such a way? .... I have said that sensual pleasures are of little satisfaction, much stress, much despair, & greater drawbacks....But you, worthless man, through your own wrong grasp [of the Dhamma], have both misrepresented us as well as injuring yourself and accumulating much demerit for yourself, for that will lead to your long-term harm & suffering."
(Alagaddupama Sutta www.accesstoinsight.org/tipitaka/mn/mn.022.than.html)

Perhaps the following Sutta also sheds some light on how the diversity, in its negative aspects, could come about:
"Monks, these two slander the Tathagata. Which two? He who explains a discourse whose meaning needs to be inferred as one whose meaning has already been fully drawn out. And he who explains a discourse whose meaning has already been fully drawn out as one whose meaning needs to be inferred. These are two who slander the Tathagata."
(Neyyatha Sutta www.accesstoinsight.org/tipitaka/an/an02/an02.025.than.html).

As to the issue of Sangha unity, it may be recalled that it had occurred already during the time of the Buddha when Devadatta attempted the schism, based on his Vinaya innovation, and the monastic code thereby prohibits causing of the splitting.

Itivuttaka 18: "When the Sangha is split, there are arguments with one another, there is abuse of one another, ganging up on one another, abandoning of one another. Then those with little confidence [in the teaching] lose all confidence, while some of those who are confident become otherwise." Doomed for an aeon to deprivation, to hell: one who has split the Sangha..."

As the Buddhism spread further throughout the linguistically and culturally diverse regions of the Far East, South East and Central Asia, if the Buddhism is to be woven into their respected cultures, some adaptation might have been considered--e.g. in regions where the cold climate might discourage the sight of robes made of cast-off rags or the idea of revering solitary sramana and homeless/jobless life style.

In the early period the oral transmission was the primary means. Perhaps it may be considered more or less inevitable that some variations might gradually accumulate in the long run merely in its technical considerations--as can be witnessed in the game of so-called "Chinese Whispers".

Ani Sutta:
"in the course of the future there will be monks who won't listen when discourses that are words of the Tathagata — deep, deep in their meaning, transcendent, connected with emptiness — are being recited. They won't lend ear....But they will listen when discourses that are literary works"

All in all, the diversity, in its negative forms, might be taken to be inevitable when considering the imperfection--both technical and moral--of human capacity. In fact, the Buddha had already predicted that the Sangha would last for 1000 to 500 years (Cullavagga). But its considerations might also be balanced together with positive aspects such as the effect of maximizing the Sangha admission and Dhamma dissemination in number and in space, in exchange for durational stability in time.

Kimila Sutta:
"Kimila, there is the case where, when a Tathagata has become totally unbound, the monks, nuns, male lay followers, & female lay followers live without respect, without deference, for the Teacher; live without respect, without deference, for the Dhamma... the Sangha... the Training... concentration... heedfulness; live without respect, without deference, for hospitality. This is the cause, this is the reason why, when a Tathagata has become totally unbound, the true Dhamma does not last a long time."

Saddhammapatirupaka Sutta:
"[T]here is the disappearance of the true Dhamma when a counterfeit of the true Dhamma has arisen in the world .... [when] five downward-leading qualities tend to the confusion and disappearance of the true Dhamma. Which five? There is the case where the monks, nuns, male lay followers, & female lay followers live without respect, without deference, for the Teacher.... for the Dhamma… for the Saṅgha… for the training… for concentration."

Certain academic rendering of the accounts might be inclined to explain away altogether all matters of conflicts and differences solely in terms of socio-political motives, sectarian ego, or sentimentalities, and while those factors are indeed significant and more or less play significant role at times, they may not describe the core and the whole of the issues at hand. It is recalled that the Buddha described the diversity of the views of other schools in the simile of blind men touching different parts of one single elephant (the whole Truth).

Tittha Sutta:
"...the man showed the blind people an elephant. To some of the blind people he showed the elephant's head, saying, 'This, blind people, is what an elephant is like.' To some of them he showed the elephant's ear, saying, 'This, blind people, is what an elephant is like.' To some of them he showed the elephant's tusk... the elephant's trunk... the elephant's body... the elephant's foot... the elephant's hindquarters... the elephant's tail... the tuft at the end of the elephant's tail, saying, 'This, blind people, is what an elephant is like.'....not knowing what is Dhamma and what is non-Dhamma, they keep on arguing, quarreling, & disputing, wounding one another with weapons of the mouth, saying, 'The Dhamma is like this, it's not like that. The Dhamma's not like that, it's like this.'"

Right after his Enlightenment the Buddha initially hesitated the dissemination because he expected only a very few could possibly understand the teaching:

Ayacana Sutta:
"This Dhamma that I have attained is deep, hard to see..."

In the absence of Sammasambuddha in the present era, perhaps the simile of the blind men may even be applicable, despite to a lesser extent, to the history of Buddhism itself. After all, the only one who is capable of grasping the whole of the elephant is the Sammasambuddha, and in this regard there may possibly be cases when even Arahants or Paccekabuddhas may not be able to fully explain to us the issues at hand.

Ultimately, then, can we finish off by saying that all is due to our existential condition, human failing? But the aforementioned Saddhammapatirupaka Sutta goes on to say that the efforts in their inverse "tend to the stability, the non-confusion, the non-disappearance of the true Dhamma," and perhaps the entire Buddhist history is ultimately about the story of human struggle to continue, preserve, and maintain the Wheel of Dhamma in motion while enriching the tradition amidst this ever changing world. In order to benefit mankind and spread the teaching in spite of our imperfection, the accumulative effort in correct adaptation, careful application, right interpretation, and accurate translation must also be met by the subtractive effort in doctrinal discernment, correction, and purification as well as preservative effort in maintaining, conserving, and preserving the tradition in respect and in humility. In my opinion, the Abhidharma schools examined in the course represent the process of these very efforts--despite some disagreements amongst each other, the Abhidarmikas, in their own disparate ways, in their own situations, had been, in their own sincere efforts, applying the teaching of The 4 Right Efforts.

Much thanks to our predecessors, the core aspects of the doctrines, in my sincere observation, seem all identical and the essential aspects of the practices seem to be the same; what we have in our possession today have withstood centuries of the test of time, and many of the differences in teachings can ironically be very insightful in allowing to see different aspects of the inner/underlying meanings while not allowing to grasp merely their external forms.

Thank you, I think those are

Thank you, I think those are great questions--probably with long and various replies. That inquiry must also have been on the background of the doctrinal debates and development flourished during the Abhidharma period. The first one especially is very much related to the subject of our course--and so is the second one,-- and everyone is welcome to share your thoughts, with references. If you're interested, you should develop your own response as a topic for your essay.