I am talking about historians, not people participating in this thread. Find me one major historian who doesn't acknowledge that the Byzantines were a separate entity. Sure they came from the Romans, but they became their own thing... you know, like the HRE became separate parts. This isn't a hard thing to grasp, but some of you seem to want to argue against accepted historical concepts... and I am not doing it.
Well undeniably at the time of the fall of Constantinople to the Turks, the Byzantines were vastly different from the Roman Empire. But 1000 years had elapsed. If the Western Empire didn't fall, surely it wouldn't have looked the same in 476 as it would have in 1476.
I know you argue that the start of the Byzantine Empire is 330 when Constantine founded Constantinople. I strongly disagree with this date as there was no fundamental differences at that time with Rome. They still spoke Latin, worshiped Jupiter and had the same culture and customs of Rome.
While that was the start of the transition from Roman to Byzantine, and you can clearly trace the roots of the future Byzantine Empire to that date, there wasn't a clean division at that time. Prior to that Rome wasn't even the capital of the Empire, Ravenna was.
As the center of power shifted to the East, Christianity became the state religion, Greek replaced Latin and all the customs of the old Republic/Empire died off, then there is a clear shift. However that date is tricky and up for debate, even among historians.
Either way the Byzantines were pretty awesome, they were the most educated people of the Middle Ages, had female doctors, high literacy, a form of central heating etc. Nothing like the 1000 years of decline portrayed by Gibbon.