Other posts on this subreddit make the case that because the Byzantine Empire was a straight continuation of Rome, Rome cannot be said to have fallen. Is this the accepted view among historians? Was the Byzantine Empire just Rome with its capital moved? If not, when did the Eastern Roman Empire "become" the Byzantine Empire? How different were the Byzantine Empire's culture and institutions compared to Rome? What's some good literature on the topic?
The term Byzantine Empire was never actually used by contemporaries - it was first coined in the 17th century by German historians (when Germany, or the Holy Roman Empire as it was then known, claimed themselves as the straight continuation of Rome). As for whether "Rome cannot be said to have fallen" - well, Rome as a city had fallen, to the Goths, Vandals, Lombards - and the Western Roman Empire ceased to exist. In the East, however, the Roman Empire was said to prevail; they called themselves "oi Rhomaioi", the Turks called them "Rum", and the western Europeans called them "Romania".
Their culture had its similarities and differences. They spoke Greek (Medieval Greek casually, the Greek of Classical Athens in the context of the court), but this was seen as the language of the Roman Empire, as it had been in the Roman East for centuries. They were also, of course, Christian, which had been their religion since Constantine the Great had first split the Empire.
Their economy was very similar to that of Rome, depending on a gold standard (the solidus) which remained very much constant until the Empire's decline, and it depended on Mediterranean trade, which their capital of Constantinople was in an excellent position to exploit. Their military largely depended on mercenaries - very good mercenaries. These included the Varangian Guard, a squad of Vikings which were sworn to protect the Emperor, the Almughavars from Spain, and cavalry from all over Anatolia, most of which was heavily armoured and armed with either lances or arrows. They also had a very powerful navy which was essential to the Empire's defence. They were culturally very Greco-Roman, preserving the practises of toga-wearing and chariot-racing, for instance, but they gradually drew more upon fashions of the Near East, with Emperors wearing turbans and slippers by the medieval period.
As for when they became distinct from the Roman Empire they succeeded, this essentially begins with their wars against the Muslims. With Islamic expansion came a massive loss of territory and, eventually, their destruction, as well as a sense that Constantinople was an essential force for the defence of Christendom. At around the same time, the Roman legionary tradition also disappears to make way for an Imperial army more suited to the Medieval era.
To learn more, there is a fantastic recent book called Byzantium by Judith Herrin, which is where I've got most of the facts for this post.