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In his book ,” Sarvastivada Abhidharma “ Bhikku K-L Dhammajoti , at page 464 to 473 , points out the difference between the wisdom of a Buddha and of an Arhant in terms of “ traces “ ( vāsanā ) , which are the subtle drives or impulses , left behind , after the defilement have been abandoned and by consequence clarify the meaning of Buddhahood state .
A Buddha superior wisdom is the result of the complete purification of all the defilments , together with their vāsanā . They are of two types : the first are abandoned by insight and the second by cultivation . The later , even though , it has already been purified by an Arhant , it may or not may manifest , depending on the faculty of the Arhant and their acts resembling defilments are involuntary , and therefore morally non-defined , because they are due to the conditioning of the vāsanā .
A Buddha state , is the result of a complete purification of all obscurations and the corresponding vāsanā , which never manifest again and the omniscience is obtained ( perfect knowledge ) . This omniscience is a state of perfect wisdom , representing a powerful force which arises at the time of attaining the supreme perfect enlightenment ( anuttarā samyak sambodhi ) , counteracting the vāsanā and the non-defiled ignorance . The Sarvastivada doctrine , sustain that the abandonment of defilments is a fruit of knowledge , which dispels ignorance , on account of which , beings are continuously bonded to Samsara . Ignorance is of two types : the “defiled “, or the cognitive basis for defiled actions and “ non-defiled “ which performs actions although involuntary and morally non-defiled , but still represents obscurations which prevent the totalistic vision of all knowable. So , purification of vāsanā and non-defiled ignorance are directly related to the attainment of Buddhahood .
In the Indo-Tibetan Buddhism tradition , the word Buddha has been rendered into “ Sangye “ . This has two parts : first “ San “ means completely purified from the two obscurations ; the first is the obscuration of the ignorance connected to the self-grasping-self and the believe of the intrinsic existence of the self and of all phenomenas , also called cognitive knowledge . The second is called the omniscience obscuration , which prevent to perceive all what it needs to be known . “ Gye” mean , blossomed , because as a consequence of the complete purification of the mind, the self-awareness clearly blossom with the nature of the primordial wisdom and one is , as awaked from the sleep of ignorance .
In my opinion this description is connected with the Sarvastivada descriptionof the Buddhahood state, which is quite different from that of the Arhant .
Warm regards
Antonio

Following up from my previous reply, this question leads us to examine the relationship between Theravada and Sthaviravada.

While it's true that the Sanskrit word "sthavira" corresponds to "thera" in Pali and that they are sometimes used synonymously, let's examine whether there are any significant differences in points of view or contexts from which the terms were used, as well as any semantic shifts that might have occurred in history--after all, the Buddha also taught us to distinguish questions that "should be answered categorically [straightforwardly yes, no, this, that]", and those "that should be answered with an analytical (qualified) answer" (Pañha Sutta AN 4.42). In that process, we might also encounter the usages of other labels such as "Vibhajjavada", "Nikaya Buddhism", "Early Buddhism", "Haimavada" (雪山部), or "Hinayana".

On Sthaviravada
-----------------------

It appears to be not an uncommon practice to roughly equate the two terms out of convenience. For example, one book reads, "'Theravada’ (Sthaviravada) of the Pali ‘Canon’" and "Sthaviravada (Theravada)"--as if they are synonymous when taken out of context.
(p. x, Indian Buddhism, by Warder)

Another book reads: "It is not possible to say if Asoka belonged to any particular sect either of the Theravada or of the Mahiisamghikas"--evidently this book does not use the word "Sthaviravada" but instead "Theravada" is used exclusively throughout.
(p. 427, Early Buddhism and its Origins, by Vlshwanath Prasad Varma)

Another example: "Sarvastivada was one of the major schools which arose about this time; as distinct from the Mahasanghikas, the Sthaviras and the Pudgalavadins," and later, "...this is a Theravadin (Sthaviravadin)"
(The First Argument for Sarvastivada, by David Bastow)
http://buddhism.lib.ntu.edu.tw/FULLTEXT/JR-ADM/bastow.htm

But it has been observed that, strictly speaking, "Theravada tradition cannot be regarded as entirely representative of the Sthavira historically."
(p. 560, An Introductory Dictionary of Theology and Religious Studies, edited by Orlando O. Espín, James B. Nickoloff)

Here's an explanation of one Theravadin scholar who agrees with making that distinction: "The term sthavira (mean­ing ‘el­der’) is the Sanskrit ver­sion of the term bet­ter known to­day in its Pali ver­sion thera, as in Theravāda, the ‘Teaching of the Elders’. The orig­i­nal Sthaviras, how­ever, are by no means iden­ti­cal with the mod­ern school called Theravāda. Rather, the Sthaviras are the an­ces­tor of a group of re­lated schools, one of which is the Theravāda."
(p. 4 Why Devadatta Was No Saint, Bhikkhu Sujato)
http://santifm.org/santipada/wp-content/uploads/2012/10/WhyDevadattaWasN...

Even thought there definitely is a close affinity between Sthaviravada and Theravada, the former in other historical context also appears to be closely associated with Sarvastivada: "At least three schools claim they possess the original Vinaya, called the Vinaya of Kasyapa or af Upali, compiled at the first council: the Mahisanghikas, Sthaviras (or Sarvastivadins?) and the Vitsiputriyas." (p. 171). The author obviously reserves the judgment but indicates their close association: "Vinaya of the Sthaviras (or rather of the Sarvistivadins)" (p. 172). Sammatiya "accepted as the forerunners not only the Sthaviras and Mahasangghikas but also the Vibhajyavadins" (p. 536). It also adds that, according to some source, "the Vibhajyavadins are Sthaviras and not Mahisanghikas"
(p. 546 History of Indian Buddhism by Lamotte)

Here's from another book: "An alternative name of the Sthaviravadins is given as Vibhajyavadins. It is doubtful whether there was any independent school having the name of Vibhajyavada. It has been shown above ... that Vibhajyavada was sometimes affixed to the name of a school on account of certain adherents differing in minor points from the principal doctrines of a particular school and preferring to distinguish themselves as Vibhajyavadins of that particular school. In this way, we may explain the Vibhajyavada of the Ceylonese tradition, that is the Ceylonese did not accept in toto the doctrines of Theravada and preferred to distinguish themselves as Sthavira-vibhajjavadi or simply as Vibhajjavadi.In the Kathavatthu, the term Sakavada is used instead of Sthaviravada or Vibhajjavada."
(p. 211 Buddhist Sects in India by Dutt)

The inquiry thus leads to ask what exactly was the thing they were referring to when they said "Vibhajavada".

On Vibhajavada
----------------------

The term "Vibhajavada" (Teaching of Analysis, 分別說部) is commonly identified with Theravada, and Hetuvada (school of causality, 說因派), Ekamsavada (school of definite/categorical answer), or Yuktivada (school of logic/reason) with Sarvastivada. One dictionary defines the word "Vibhajjavada" as follows:
"The name given to the Dhamma by the orthodox; the term is identical with Theravada and the Buddha is described as Vibhajjavadi."
(p. 889 Dictionary of Pali Proper Names, Volume 1 by G.P. Malalasekera)

One particular book explains thus: "The Theravada arose from the Vibhajyavada, which was a school opposing the Sarvastivada."
(p. 512 Awakening of the Heart: Essential Buddhist Sutras and Commentaries by Thich Nhat Hanh)

That designation appears to have held a particular function of characterizing a certain doctrinal emphasis in contradistinction to that of other school which they viewed as being heretical. So it appears that, in a particular event in history, Sarvastivada was called "Ekamsavada" for distinguishing a relevant doctrinal aspect--categorical answer or singular point of view--whereas other times it was called by other names--Hetuvada or Yuktivada--to highlight its logical and analytical character of its Dharma theory. The designation Vibhajjavada, then, is used to emphasize its analytical (conditional, qualitative) aspect of the doctrine in contradistinguishing with the Ekamsavada (absolute, categorical) aspect. If so, then it would perhaps be somewhat awkward if we were to pull out the terms out of context and compare "Vibhajjavada" alongside or in contrast with "Yukivada", even if they more or less refer to Theravada and Sarvastivada respectively--just my observation.

"There is no distinction in the chronicles between the Theravada and Vibhajjavada; but Theravada is probably a general term meaning merely 'authentic doctrine'. The word is used in this sense also with refence to the schools of Buddha's predecessors .... The Vibhajjavadins declared that they taught the Theravada. However, the same thing is also maintained by other sects, such as the Mahissasakas and the Sarvastivadins.... In Majjh. II, 197(99) Buddha says referring to himself that he is a Vibhajjavada, i.e., 'one who explains everything with careful distinctions,' and not an ekamsavada, 'who answers questions from one point of view only'.
(p. 8 A History of Indian Literature by Moriz Winternitz)

"While the Sarvastivadins were not much represented in south India, this was an area where the Vibhajjavadins, or 'Distinctionists', were strong. The fraternity which became established in Sri Lanka, which called itself the Theravada (Pali equivalent of Sanskrit Sthaviravada) followed this school of thought. Theravada survived in the south till the seventeenth century, and then withdrew to its stronghold in Sri Lanka."
(p. 86 An Introduction to Buddhism, by Peter Harvey)

Another dictionary explains thus: "VIBHAJJAVADA. An obscure sect named 'The Distinctionists' that belonged to the Sthavira (elder) tradition. The name of the sect can probably be traced to the Buddha's practice of making distinctions between extremes. At the Council of Pataliputra in about 250 CE, they were declared to represent the orthodox teachings of the Buddha in opposition to the heretical position of the Sarvastivadins."
(p. 238 Historical Dictionary of Buddhism, by Carl Olson)

A Theravadin scholar explains thus: "In later times the Buddha's teaching, as documented in the Paali Tipi.taka and handed down by the Theravaada school, was called vibhajja-vaada, i.e., the discriminative, differentiating, analytical or critical teaching, in contrast to a generalizing and one-sided (eka.msa) doctrine."
(Anguttara Nikaya by Nyanaponika Thera)
http://www.accesstoinsight.org/lib/authors/nyanaponika/wheel238.html

Another book describes other point of view: "In the Sarnath inscription, it is stated that the Sarvastivadis ousted the Theravadins there...."
The subsequent paragraph describes Vibhajyavada: "it is noteworthy that they are not uniformly recorded by the traditions as a distinct sect nor any considerable period of time," and "according to the Sammatiya tradition, Vibhajyavada like Sankrantivada, developed from the Sarvastivada sect, the Mahasanghika tradition (second list of Bhavya) would truncate the early Buddhism into three schools i.e., the Sthavira, the Mahasanghika and the Vibhajyavada...," and "at the time of the third Buddhist Council, all true Buddhists are described as Vibhajyavadins."
However, it adds that "Ii]t was possibly due to their analytical attitude within the general framework of the Sarvastivada doctrine that they got the name of Vibhajyavadins or Sarvastivada-Vibhajyavadins." (p. 51)
But on other hand, "[a]n alternative name that is sometimes given to Theravada is Vibhajyavada," and adds that, "it is difficult to agree with the opinion that the Ceylonese theravada was a late derivateive from the original Theravada. In fact, the Ceylonese Theravada appears to be a very ancient school and reflects the Mula-Sthaviravada tradition to a remarkable degree.
(p. 53 Buddhist Sects and Sectarianism by Bibhuti Baruah)

It gets to be confusing. We might be missing some other sources too. But I think we could grasp a rough outline from the following description in Encyclopedia of Britannica:
"The first separation occurred in the 4th century bce at the time of the second council, when a group broke away from the Sthaviravadins and became known as the Mahasanghika. A second major schism occurred when the Sarvastivadins (who hold that “all is real”) separated from the Vibhajyavadins (adherents of the Doctrine of Distinctions), presumably the Sthaviravadins. Those Vibhajyavadins that were distributed geographically in southern India and Sri Lanka became known as the Theravadins (the Pali form of Sthaviravadins)."
https://www.britannica.com/topic/Theravada

The factors that should be taken into account, then, are how a particular term had been used for what particular function, in which community, in which period in history, to emphasize what, in contrast to what, from which point of view, and in which context, background, and aspect of the Dhamma. After all, even the corresponding word for "buddhism", as we use the term today, was not even present or invented in India when the Buddha was alive--he may have used the word "dhamma" or other terms instead. During that time, there might not even be a corresponding Indian word for "religion", as is used in our modern academic sense, either.

By replying this way, I end up choosing the non-categorical approach. Obviously, non-categorical answer do not necessarily offers us an extreme stance of naive relativism, as it were a license to deconstruct every labeling into ambiguity and to reconstruct arbitrary semantic cuts for subjective preference. The historical designations instead reveal us many aspects of the significant issues that may arise in the depth and subtlety of the doctrine of the Buddha as sammādiṭṭhi. As we attempt to grasp the dynamics of semantic shifts, expansions, contractions, and synthesis that might have occurred throughout the long history of Buddhism, perhaps the following remark made by Benjamin Lee Whorf, a linguist, can be appreciated:

"We dissect nature along lines laid down by our native languages. The categories and types that we isolate from the world of phenomena we do not find there because they stare every observer in the face; on the contrary, the world is presented in a kaleidoscopic flux of impressions which has to be organized by our minds - and this means largely by the linguistic systems in our minds. We cut nature up, organize it into concepts, and ascribe significances as we do, largely because we are parties to an agreement to organize it in this way - an agreement that holds throughout our speech community and is codified in the patterns of our language. The agreement is, of course, an implicit and unstated one..."
(p. 213 Science and Linguistics)

If we were to apply this remark to the history of Buddhism as a multi-dimensional phenomenon, perhaps it challenges us to consider not only the possibility of implicit speech communities of the bygone eras, but also the limits of our own communities and of the provisional nature of academic consensus.

----------------------
Finally, I'd like to roughly lay out what Akira Hirakawa had written on the subject in his book, A History of Indian Buddhism: From Sakyamuni to Early Mahayana. Please take it as you will and share your opinion with us.

According to the Preface:
"Hirakawa's history maintains a better balance and is more comprehensive than many English-language histories. Earlier surveys of Indian Buddhism have generally emphasized either one aspect of Buddhism, such as Theravada, or one approach, such as Buddhist philosophy. Hirakawa's history includes three types of discussions: historical, bibliographical, and doctrinal.
(p. x)

"The early order eventually divided into two schools: the progressive Mahasanghika and the conservative Sthaviravada (P. Theravada).
(p. 2)
"the Theravada, Sarvastivada, Sautrantika, Sammatlya (all of Sthaviravada lineage)"
(p. 2)
"About one century after the Buddha's death, the early order split into two schools, the Sthaviravadins and the Mahasanghikas. These schools subsequently suffered schisms that eventually resulted in at least eighteen schools."
(p. 70)
"The Sutra-pitaka of that canon was divided into five collections (Nikiiya), a format maintained by the Vibhaijavadins of the Theravada tradition.
(p. 71)
"...and the Sthaviravada (P. Theravada)"
(p. 81)
"the basic schism ... resulted in the formation of two schools: the Mahasanghika, whose monks refused to accept the conservative ruling of the committee of eight monks, and the Sthaviravada (P. Theravada)"
(p. 81)
"the Mahasanghika, whose monks refused to accept the conservative ruling of the committee of eight monks, and the Sthaviravada (P. Theravada), whose monks agreed with the conservative ruling.
(p. 81)
"the term "Nikaya Buddhism" refers to monastic Buddhism after the initial schism into the Mahasanghika and Sthavira schools had occurred. It must be remembered, however, that other groups of Buddhists existed at this time."
(p. 105)
"Thus, according to the Sri Lankan tradition, after the initial schism the Theravada and Mahasanghika schools each held a separate council.
(p. 108)
"According to the Dipavamsa account, the monks of the Great Assembly compiled new versions of the sutras and vinaya quite different from those of the Sthaviras. This group is called "the monks of the Great Council" (Mahasangitika) in the Dipavamsa and "the Great Assembly" (Mahasanghika) in the Mahavamsa. The name "Mahasanghika" meant that these monks constituted the majority of monks at the initial schism. Thus, according to the Sri Lankan tradition, after the initial schism the Theravada and Mahasanghika schools each held a separate council.
(p. 108)
"The Theravada School was also called the Vibhajjavadin (those who discriminate) School.
(p. 110)
"divisions in the Sthavira lineage began occurring during the third century after the Buddha's death. First, the Sarvastivadin (also known as the Hetuvada) School split away from the Sthavira (or Haimavata) School."
(p. 110)
"The fate of the original school of the Sthaviras is not so clear. The first schism in the Sthavira lineage
resulted in the Sarvastivadin and Haimavata schools. Although the Haimavata School is called the "original Sthavira School" in the Samaya, the Haimavata School was influential only in an area in the north and was far from central India, where most of the important events in very early Buddhist history occurred. Moreover, the school does not seem to have been very powerful. The other schools in the Sthavira lineage split off from the Sarvastivadin School. Consequently, the account found in the Samaya seems questionable. Vasumitra, the author, was a Sarvastivadin monk, and may have written this account to demonstrate that the Sarvastivadin School was the most important school among those in the Sarvastivadin lineage. Vasumitra's overall position thus would seem to conflict with his statement that the Haimavata was the original Sthavira School."
(p. 111)
"According to the Sri Lankan chronicles, the schisms in both the Mahasarighika and Theravada (Sthavira) lineages"
(p. 112)
"The later schisms in the schools of the Theravada (Sthavira) lineage begin with the formation of the Mahirpsasaka (Mahisasaka) and Vajjiputtaka (Vatsiputriya) schools out of the Theravada School."
(p. 114)
"In the Samaya, the Haimavata is identified with the Sthavira School formed at the time of the initial schism, and is thus one of the oldest schools. In the Mahavamsa, in contrast, it is listed as a later school."
(p. 114)
"the Sthavira School was influential in the western and northern parts of India, while the Mahasanghika School was dominant in the central and southern parts of India."
(p. 120)
"The names of many schools belonging to the Sthavira tradition, such as the Sarvastivadin, Theravada, and Sammatiya..."
(p. 120)
"the Abhayagiri sect maintained close relations with Indian Buddhists and adopted many new teachings from India. In contrast, the Mahavihara sect has carefully maintained the Vibhajjavada tradition of Theravada Buddhism until the present day.
(p. 124-125)