Not at all. There was direct continuity here -- the Romans remained Romans in their eyes and never stopped. Indeed, the capital of the Empire never changed between 324 (well within what you'd call the Roman Empire) and 1204 (well into what you'd call the Byzantine Empire, when it was sacked and occupied by the Fourth Crusade).

The issue is that you are viewing a single entity differently, because of a hack in the way history was taught, biased by a Western Catholic church-based perspective.

The nature of being Roman changed from the early days of Rome to the final fall of the last remaining bastions in Crimea in the 1470s, of course. But the Republic was different to the early Empire, was different to the Empire in the 3rd century, was very different to the Empire in the 4th century, was different again to the Empire in the 6th century, was different to the Empire in the 9th and 10th centuries, etc, etc.

But that's no different to me pointing out that the way people thought and lived and worked in the United States in the late eighteenth century was very different to how they live now in the 21st century. But we still consider these people to be American.