Hey! I'm still working on my degree in history, so more established historians please criticise me as I imagine it will help with my dissertation aha! I also not too good at formatting so my referencing may be a little sloppy.

The short answer is, We as historians use the term "Byzantine" for clarity. As it makes it easier to distinguish between the two periods of Roman History and makes sourcing much easier. As in every way, the Byzantine Empire changed as time progressed, from the social landscape to the political one. But I'll also explain how the Byzantines thought themselves "Roman", but why it eventually changed.

As the Eastern Roman Empire survived much later than its western counterpart, and as such, the differences were great. Culturally, The Byzantines preserved much of the Romano-Hellenistic culture of the later Roman Empire, however, it progressively got more and more "Greek" as time progressed. You are correct in stating that the Byzantine Empire felt themselves to Roman, Richard A. Gabriel states in The Great Armies of Antiquity

"Byzantines never thought of themselves as anything but the continuation of the Roman Empire" 277

Historian Demetrius John Georgacas explains in his work that Constantinople was referred to as Nova Roma, or "New Rome". ( See The Names of Constantinople pages 347-367 of Transactions and Proceedings of the American Philological Association) . Thus, we know it's clear that the Byzantines felt themselves to be Roman, but oftentimes Citizens of Byzantium would refer to themselves as Byzantine, such as in Anna Komnene's The Alexiad. By it wasn't a "catch-all" term for the empire until much later.

Well, the reason the term changed was when German Historian Hieronymus Wolf Published his collection of sources titled; Corpus Historie Byzantine. Wolf coined the label "Byzantine" to distinguish the later parts of the empire, for what reason we can only speculate. I have seen Historians theorise that it's due to the fact Wolf was a member of the Holy Roman Empire, which posed itself as the second Rome. And over time, it just stuck as an easy way to distinguish the two parts of the empire's lifespan.

The last part of your question is interesting to me, do we consider it a continuation of the Roman Empire? As we know, the Byzantines were catholic from their inception, and geographically it didn't even hold Italy. Thus culturally it would have evolved differently, taking more influence from its eastern neighbours, and holding more fertile and prosperous lands, such as Egypt, which promoted economic growth. To me, I see it as a Ship of Thesus' kind of idea. If you replace every aspect of the empire with something new, is it still the same empire? Now obviously the Byzantines carried on ideas from the Roman side, a big one being the laws. Justinian I codified many laws of the past Roman Jurists within the Corpus Juris Civilis. So there were still aspects of the traditional Roman Empire remaining, but as time passed it naturally evolved. Thus by continuing to refer to the Byzantines as such, we make it much easier to discuss the empire and draw distinctions vital for study.

I hope I answered the question well and met the standards of r/AskHistorians, I'm a little nervous about posting ahaha

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Edit: Forgot to list sources I used lmao, they're all mentioned but I'll list them below for ease.

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Primary Sources:

The Alexiad- Anna Komnene

Corpus Juris Civilis- Justinian I

Secondary Sources:

The Great Armies of Antiquity- Richard A. Gabriel

The Names of Constantinople- Demetrius John Georgacas

Corpus Historie Byzantine- Hieronymus Wolf