Always and never, depending on who you talk to! For myself, I don't think it's a problem for people to use whatever term they want, as long as they are aware of the nuances involved.

Periodisation is complicated and it is difficult to say when something began and when it ended. Yes, many Byzantinists write books about Byzantium from the time of Constantine I (fourth century), Justinian I (sixth century), or the crises of seventh century, but others are reluctant to use the term at all, preferring to use 'Roman' to refer to the empire. In recent years historians are also increasingly aware of the theoretical problems involved when we impose broad categories on the past. Can we describe an empire with an extensive history simply as being 'Roman' or 'Byzantine'? Are we imposing our ideas about a 'nation-state' or a 'multi-ethnic empire' on the past? I don't have the answers to these questions, since scholars are still debating these issues even now. But I would like to argue that for non-historians, it is absolutely fine to refer to the empire as either 'Roman' or' Byzantine' when talking about the late antique and medieval polity ruled by Constantinople, so long as they are aware of how artificial these terms are when we talk about the past.

This might seem like a terrible answer, but I think it's the best I can do right now. What were the turning-points? The separation between the east and west Roman empires? This is highly debatable, since we can question how separated the 'two empires' were in reality, as I did here. Or maybe something changed in the sixth century, when the last native Latin-speaking emperor launched his ill-fated wars of reconquest in the west, provinces that were later marginalised and of little use to an increasingly eastern empire? Hardly, Justinian and successors saw themselves as Romans, and new research has uncovered more and more the important roles played by both western provinces and indeed territories beyond imperial borders in the imperial world-view. For many, the Roman world had not been shattered in the sixth century, but had instead endured.

The seventh century is often noted as a century of crisis, which ushered in a 'Dark Age' for the empire, according to some historians marking a suitable point for the 'Byzantine' phase of the empire. Once again, this view has been largely overturned, as we no longer see this period as a time when the empire was in total decline, nor a time when we lack the sources to write an interesting history. Yes, there were changes, but for every changing institution, there was a continuation of another. So what if the senate disappeared, when the aristocracy now served the emperor in different ways (as I argued here)?

Still, it is true that many historians do hold the seventh century to be a convenient way to delineate between the 'Roman' and the 'Byzantine' phases of the empire, but there is always a fundamental understanding that it was the same empire and that much had lived on even when the polity is described with a different term. This is perfectly fine, because historians are (generally) aware of the important continuities between them, making this only a term of convenience to point out some of the differences. However, it is important to keep in mind that the empire was so much more than just ruling provinces or maintaining its military superiority. Indeed, historians are now even interested in the world beyond the empire and how it interacted with and was influenced by the empire. I don't know how to express this properly, but maybe it is just a case of remembering that it was always the same polity, an empire founded by Augustus and was always recognised by contemporaries as such, as well as to think critically every time someone use the word 'Byzantine' - what do they mean by that word? How does that shape their writings? It's hard to remember these nuances when reading about the empire more generally, but if you keep this in mind, I think you will do more than fine.

If anyone is interested in further reading, I'll mention some of my sources as well. For a neat summary of recent problems in 'Byzantine' studies, particularly in how it relates to other fields and the problem of definitions, see A. Cameron, 'Late Antiquity and Byzantium: An Identity Problem', Byzantine and Modern Greek Studies, 40.1 (2016). Another useful survey of the historiographical issues is Cameron's Byzantine Matters (2014). To get a sense of how complicated 'Roman'/'Byzantine' identity was, see I. Stouraitis, 'Roman Identity in Byzantium: A Critical Approach', Byzantinische Zeitschrift, 107.1 (2014). For the argument of those who argue for Byzantium's perpetual Romanness, the most articulate voice is no doubt Anthony Kaldellis, and his recent The Byzantine Republic: People and Power in New Rome (2015) provides a fine overview of his argument, though in the near future he will publish a book specifically on Roman identity, so keep an eye out for that. For my own interests, the transformation(s) experienced by the empire in the seventh century, J. Haldon, The Empire That Would Not Die: The Paradox of Eastern Roman Survival, 640–740 (2016), is by far the most up-to-date volume on the issues involved.