The Byzantine Empire is the Roman Empire. I know it seems weird! They were Christian, they spoke Greek, they only controlled the eastern half of the old empire and kept shrinking until there was nothing left...but for an empire that lasted 1500 years, of course a lot of things changed.

When Constantinople was conquered in 1453, it was very different from Rome in 27 BC. But "Rome" the city and "Rome" the empire aren't the same thing. The empire changed a lot by the year 100, 200, 300...eventually it was split into a western and eastern half with two different emperors ruling together. One ruled in Constantinople and the other ruled in Ravenna, or Milan, or somewhere else, not even the city of Rome. So when are the changes too different? When did "Rome" stop being "Rome"?

The reason we call it the "Byzantine Empire" now has more to do with modern historians and western European prejudices against eastern Europe. For that, see u/AksiBashi's recent answer about when the term Byzantine empire was introduced

I also wrote about what other people in Europe called the Byzantine Empire

But as far as people living in the Byzantine Empire were concerned, the Roman Empire was still alive and well and they were Roman citizens.