Traditionally in the West, part of the differentiation is in language and the eventual split in religion - Latin vs. Greek - Catholic vs. Orthodox. The tail end of the Roman Empire and its western successor was Latin and Catholic while the Byzantines, the eastern successor was Greek and Orthodox.

The Byzantines in their minds never ceased to be Rome and be nothing other than Roman

Well to some of us the fall of Constantinople WAS the end of the last vestige of Rome. Clear and obvious. But I understand why others can't or don't understand and agree with that. Its also easier to differentiate between Byzantium and 'other' so called Roman Empires (Holy ones) by using the B name, especially when Rome itself was long out of its possession.

See this thead and a couple of other ones you get if you search the sub. As been rehashed many times, the "fall of the Roman Empire" wasn't an event but a process of receding territorial claims, political power in Rome, etc. But in the end, it was a clear, dramatic shift in the political structure that resulted in the loss of a capital, most of the land, and a reputation. That's enough to call it a "fall," even if an emperor was still ruling under the same name.

After the coronation of Charlemagne in 800 A.D. the West stopped recognizing rulers of Constantinople as Roman Emperors. The reasons for that are multiple and complex, but from that point on they refer to the Byzantine Empire mostly as "the Empire of the Greeks".

As for the Byzantines themselves and their eastern neighbors, there was no doubt they remained Roman. For them, Charlemagne was simply "the Emperor of the Franks".

There are multiple relevant threads in the Frequently Asked Questions, under the topics The Fall of the Western Roman Empire, Heirs to the Roman Empire, and The Byzantine Empire.