1) Why do you think Greek-speaking inhabitants of the Eastern Mediterranean continued to call themselves Romans until the early 21st century?
It may seem peculiar to think of ethnic Greeks (as well as other groups) designating themselves as Romans, or Romaioi so long after both the Roman and Byzantine empires had disappeared, but the apparent oddity of this fact is largely a product of the conventions for naming and distinguishing certain historical groups. The inhabitants of the later eastern Roman Empire are called Byzantines by us, but in their own time they would have referred to themselves, and been referred to by others, as Romans. In our perception the Byzantine Empire eventually evolved to become an entity separate from the Roman Empire which preceded and originated it, but that evolution was not problematic, or perhaps not even apparent, for those living within the empire during any of its iterations. For them it was still the Roman empire, and their culture was still Roman as well. It may seem odd that, for example, an ethnic Greek and Orthodox Christian inhabitant of Ottoman Anatolia in the 19th century could call themselves by the same name as a Latin-speaking pagan inhabitant of Italy in the classical period. It is clear, however, that while historians might neatly delineate historical periods and people groups, those delineations must often overlook the persistent continuity of many aspects of culture; a name, a religion, or even an ethos can survive long after we might suspect.
2) In what ways was the culture of the Byzantine Empire similar to that of ancient Rome?
It is somewhat challenging to answer this question without first specifying what is meant here by “ancient Rome”. We must concede the fact that we cannot effectively understand the history of the cultures in question if we choose to refer to “ancient Rome” as a single, cultural monolith. Are we speaking of the Roman Republic, the Roman empire of Augustus, or the post-Constantinian Christian empire of late antiquity? Being in its inception an equal and unified part of the late Roman Empire, the Byzantine Empire and its culture necessarily shared many fundamental characteristics with that particular phase in the evolution of the Roman state. The Byzantine Empire, like the late Roman Empire, was a predominantly Christian state encompassing a significant number of ethnic and cultural groups, and it was likewise governed by an emperor, a senate, and a hierarchy of ecclesiastical authorities. Most importantly, the Byzantine Empire was (as we discussed in the first question) in its origin a part of the larger Roman Empire, and when the west fell and the east became an independent entity it was still perceived by its members as an unbroken continuation of Roman culture. While we may discern the numerous significant differences that eventually developed between what we designate as the discrete Roman and Byzantine cultures, such differences meant little to people who held themselves to be the inheritors of the Roman name and culture, regardless of where they lived, what god they worshipped, or what language they spoke.
3) How was the Byzantine Empire different from the Roman Empire?
If the second question was intended instead to compare the Byzantine Empire with pre-Constantinian Rome, or even pre-imperial Rome, then the similarities become somewhat more scarce. Greek, the main language of the Byzantine Empire, had not succeeded in supplanting Latin in the West, despite the fact that a fluent knowledge of Greek eventually became nearly ubiquitous among the Roman upper class. Furthermore, while both the Byzantine Empire and the Roman Empire were composed of many diverse people groups, the groups which became parts of these respective empires had widely varying effects on the culture as a whole. The influences of Gallic, Germanic, and various western and southern Mediterranean cultures can easily be discerned in the western empire, while on the other hand the culture of the Byzantine Empire was uniquely shaped by the culture of Greece, the Balkans, and the Slavic lands to the north, as well various Turkish and Arabic people groups. Finally, and perhaps most significantly, the Roman Empire was for most of its existence a pagan institution, one which did not even begin to tolerate Christianity until the third century after Christ. By the time the eastern empire became independent Christianity had undergone rapid growth, eventually becoming the state religion, and one of the defining cultural elements of the Byzantine Empire.