I've always been curious about this as the justifications I've read for claiming that the Byzantine Empire was no longer Roman always seemed a bit thin - most recently I read in Imperial Tragedy, From Constantine’s Empire to the Destruction of Roman Italy AD 363-568 by Michael Kulikowski where he argues the transition to something decidedly not Roman took place at the end of Justinian's reign as the governmental structure of the empire had changed radically from its former incarnation, but it ignores the fact that the people living in the Empire at that time would almost certainly identify as Roman and not something else.
Of course the date of this contruct of non-Romanness can and will be debated forever but I'm more interested in how and why historians often make such an effort to point out the transition at all. After all, we know from many sources that for many centuries after the end of the western Empire its continuation in the East considered itself the Roman Empire and that anyone else who did so was a pretender. Even if they spoke Greek and worshipped the Christian god they still considered themselves as Romans and I would assume that needs to be taken into account.
So the crux of my question is: When did Rome stop being Roman, and how do we make that distinction when the people living at the time would likely have never even considered the concept of it? After the peace with the Sabines? After Hannibal? Augustus? When Aurelian stitched the empire back together? Constantine? The list could go on and on of course but I'm very curious about this and thank anyone who responds!
Regarding Rome, Jean Durliat claims that the germanic invations didn't modify the taxation structure of the Western Empire, so, he says, it never fell. This is debated by a lot of economic historians, such as Wickham. However, for mainstream historiography, the western half of the roman empire did fall on 476 ad. For romans, a territory would be roman if it was under roman law.
Most of the time, History is taught in a deterministic way, so when we talk about the collapse of Rome as a political entity, we use the year Odoacer deposed the last roman emperor. When Odoacer conquered Rome, he sended the Imperial Sigils to Constantinople, and thus the western half of the Roman Empire, as a political entity, ceased to exist. Having said that, while the political entity fell, that doesn't mean that "common" latinized populations stoped being romans.
We need to take into consideration that the concept of "nation" is a fairly new one, at least if we compare it to the time period we're discussing here. So this people that inhabited the roman empire were romans, and then became something else while their culture started to mix with the one of their new rullers. By all means, "bizantines" were romans, they called themselves romans, everyone around them called them romans, when the Mehmed II conquered Constantinople, he claimed he was the Kayser-i-Rûm, so, again, "bizantines" were greek speaking romans (all the eastern half of the roman empire spoke greek as a lingua franca).
Remember that the Eastern Roman Empire only started being "bizantine" in 1557 (more than a century after Constantinople fell to Mehmed), after Hieronymus Wolf created the term. Wolf was german, and lived in the HRE, a political entity that constantly tried to claim they were the true heirs of Rome. So we could argue that the "non-romanness" of the bizantines is a political claim by those who wanted to claim the prestigious "vacant" throne.
As you may imagine, your question is still being debated.