Whenever the Byzantine Empire is discussed online, inevitably someone will appear to argue against the use of that name on the basis of the Byzantine Empire really being the Roman Empire. While there's obviously truth to this, I can't help but feel as though it comes from a place of fanboyism - like the Byzantines will seem cooler if they're identified more closely with the Romans, and the discontinuities and transformations which caused historians to adopt the name "Byzantine" in the first place are downplayed. For some people, it even seems to be a matter of justice, as if the use of the name "Byzantine" is a moral slight against them. I want to understand where this comes from.
So where do actual historians of the Byzantine Empire stand on this question? Do they also tend to view Byzantine history as simply being "Roman history"? Is the Byzantine Empire seen as the single legitimate bearer of the Roman mantle, or is it portrayed as one part of a plural post-Roman Mediterranean world? Do Byzantine historians equate the terms "Roman" and "Byzantine," or even suggest that the name "Byzantine" shouldn't be used at all?
I use it. My advisor (who is one of the leading specialists of it) uses it. All the professionals use it. Even Anthony Kaldellis, the guy who wrote The Byzantine Republic who makes the argument for the continuity of Roman republican institutions all the way to 1453 uses it.
The only people who are fervent about specifically not using it have a particular axe to grind, and the axe wield doesn't go too far past the demand for the NAME to be restored.
Historians deal with periodizations, names and definitions. We also argue intensely about them. We settle on certain definitions and periodizations because they're useful, and because what we do is study these niche periods. To attempt to completely upend established precedent (which of course is always happening, and is always in the process of just being about to happen) requires the new precedent to be more useful, or else even the flawed older practice sticks around.
Take feudalism or magic. Those two words are completely loaded, highly problematic, quite outsider in construction, and... still used, despite all the caveats of its problems, its lack of formal utility. They're basically just used with "air quotes" to represent the instability of the outdated notion while still saying the notion.
Same with Byzantine. Every historian acknowledges the problematic origin of that word. So much so that I really feel anyone would be hard pressed to find a relatively general history of the Byzantine Empire without a few sentences expressing that fact.
But guess what? We use that word because it creates a specificity for us. When we say Byzantine Empire, we know we're talking about the state focused on Constantinople for much of the medieval era. If I told you I was a late byzantine specialist, you'd know that I dealt with the Palaiologan 13th-15th century. If I told you I was an early byzantine specialist you'd know that I primarily focus on the eastern roman empire of the 4th-7th century. If you call the Byzantine Empire the Roman Empire, well... now we have to divide the up the periodization. Currently when we say "Late Roman Empire" we mean from 300-600. Well if the Byzantine is now the Roman... when IS the late empire? Is it now 1261-1453? If not is it 284-1453?
Basically, how does the political act of renaming the Byzantine Empire help the scholars who NEED some kind of more concrete way to separate its later period from the earlier period, especially if doing so upends pre-existing conventions?
Short answer, it doesn't. Renaming the Byzantine Empire the Roman Empire really only serves one political purpose, and that's to reaffirm the continuity of the later form of the state with the earlier form. This purpose is quite important if you are the Byzantines themselves, or if you're playing Crusader Kings 2, where you're trying to push your claim for reuniting the Roman Empire on the Kingdom of Italy in order to have an invasion casus belli, but for scholars, it doesn't.
Another food for thought that I always like to raise as a counter example of the problems of renaming the Byzantine the Roman.
The R&B music of today has a direct continuity with the Blues music of the early 20th century. R&B stands for Rhythm and Blues, and itself was a discreet genre as early as the 1950s. However, does modern R&B like Drake sound anything like Fats Domino or Robert Johnson? Hell modern R&B doesn't even use the I-IV-V chord progression standard to blues music.
We differentiate R&B from Blues and from early 50s Rock (which was mostly blues based) because it helps us periodize and organize the common patterns of that music. To force all music from this lineage into one category of its original name, can help serve a political purpose (emphasizing the african-american origins and continuity of the music), but it would not help with the study of it without additional qualifiers, especially if changing or "enforcing" a continuity of a name goes against pre-existing conventions.
/u/B-Hooks gave an awesome answer to Is there any historiographical benefit to using the term "Byzantine" rather than "Medieval Roman"?