Byzantines were more than just a continuation of Rome. They spoke a different language, had different religions, large scale differences in government and a completely different army structure. They were definitely not the Roman Empire after the rise of Islam so that's only more than 800 years of history. Especially since Rome in civ does not do a good job of portraying the Late Roman Empire. It is all based on the Republic and early Empire. Not to mention the core of their empire was on a different continent than Rome. They were profoundly different than the Latin West or the Islamic east.
There is truth in this but the name Byzantium and Byzantine is a modern terminology. So what else do you call them, perhaps just the Eastern Roman Empire?
There was a direct line of government and legal system/code from ancient Rome to medieval Constantinople. They certainly evolved over time but that is no different from any long lived state.
In contrast the Ottoman Sultan may have called himself a Roman Emperor but in principal he was everything but the opposite.
Augustus who established the office of Roman Emperor went to great efforts to demonstrate that he was the first citizen (or the first among equals), essentially the chief public servant or the chief senator. Obviously its easy to see this was part of a propaganda campaign on his part but he knew that if he was to present his reign as a legitimate successor to the Republic he had to as Emperor place the interests of the State first in all the decisions he made, and that is what set apart Roman Emperors from the other despots of that time.
He didn't assume office with some 'mandate of heaven' or religious ideology but with an attitude of respecting the civic traditions of the State above his own self. Arguably the success of the Roman Empire was paved with these ideals, for instance Rome's five good emperors all exhibited these virtues.
Romans believed heavily in traditional family values and the Emperor was expected to follow this lifestyle (at least in public). Thus harems were frowned upon and probably seen as a aspect of the debauched oriental styles of government that would be scandalous for a Roman emperor to openly engage in.
Christianity fit in quite well with these notions - the emperor serving the interests of the state and the people and traditional family values. The Byzantine Emperors followed the same attitudes and culture to law and government. Harems were forbidden and Emperors according to church law could marry one wife (this wasn't always followed though...). Even the role of Emperor was very different to the stereotypical eastern oriental despot. Byzantine Emperors were crowned by the Patriarch and could only govern with the loyalty of the senate, the army and the people who had the Hippodrome as a public space to view their emperor.
No Byzantine emperor who lost the confidence of the army, the senate and the people could be expected to hold office. Thus the Emperor was not a sacred being, nor was he seen as Gods special representative or Gods prophet. The office of emperor was held in high esteem and with an almost sacred reverence but the actual emperor had to earn his stripes.
This attitude to government is not that far off Republican principles in certain aspects (which is where it evolved from)
Byzantium even had distinct civil law (Corpus Juris Civilis - The Code of Justinian which is still the basis to much of European law) and the church had its own distinct canon law - which governed church practice. So church and state in Byzantium may have been closely aligned in many aspects but they were also quite distinct from each other.
The Ottoman Sultans on the other hand ruled with a very different approach. There you find theocratic rule - Sharia law governed both civil and religious life, harems, The sultan was the Caliph and the 'Emperor', religious jihad wars etc. For anyone to argue that the Ottomans were a continuation of the Roman Empire is really just naive.
And i'm not even getting into the cultural, language and religious differences between the two.
Byzantium and the Ottomans had a completely different evolution in government. There is no overlap between them outside of geography and a few other aspects of their society.