Answer: aside from the fact one is speaking Latin and the other Greek, and one was polytheistic (mostly) and the other aggressively Greek Orthodox, none.....LOL
Back in 285, Diocletian realized the Roman Empire was too big for one man to run so he made an administrative East-West division with its own emperor though the older one was senior. Constantine the Great (312-337) did away with this and after the Edict of Milan, legalizing Christianity, moved the capital of the empire to Byzantium. Then Constantine, ever the modest type, renamed the city Constantinople.
After 337, the Romans went back to the old East-West division and in 476, the Western Empire fell.
September 4, 476 is the fall of the Roman Empire.
But that was only the western half.
The East lived on until 1453. It was a pretty wild ride lasting nearly a thousand years.
The word "Byzantine" did not even exist until 1557 when a German historian named Wolff coined the term to mark a distinction between the Roman Empire of antiquity and the Byzantine Empire that still carried Roman standards into battle in the Middle Ages.
The Eastern Roman Empire was built around a Greek core rather than the original Latin but it took until the 600s before Greek became the official language of the empire. Of course the character of it had changed. Historian argue weather it was Justinian or Heraclius who was the "last" of the Latin emperors but a look at the names should tell you what you need to know. LOL
Western Europe, for some reason, never seemed to recognize Byzantium as the Roman Empire. They always referred to the Emperor as "King of the Greeks." It got to the point where they crowned an aging Charlemagne as Roman Emperor in 800 and kept up the Holy Roman Empire charade until Napoleon got tired of looking at it. As Voltaire said, "It was neither Holy, Roman, nor an empire." Oddly enough, the Turks, Arabs, and Persians had no problem seeing the Byzantine Greeks as Romans, referring to the empire as "Rum".
It was all fun and games until May 29, 1453 when the Turks finally took the city by storm. Even the Turks recognized how valiant an adversary the Byzantine Greeks were and the Greeks never forgot, finally winning freedom in 1832.
Of course, they did not call themselves "Byzantines." They proudly called themselves "Greek".
