What is the difference between the Romans and Byzantines?

Although the Byzantines considered themselves to be Romans, the distinction between the two terms, “Romans” and “Byzantines” mainly derives from our historical separation of the Western and Eastern Roman Empires.
When the emperor, Diocletian, divided the Roman Empire in 286 A.D., he believed that the two sections of the empire: the West and the East, would be best ruled separately, with the two emperors working together cooperatively. Emperors who came after Diocletian disagreed and tried to reunite the two halves of the Roman Empire, but with the increasing frequency and success of barbarian invasions by the Goths, Huns, and Germanic tribes, these emperors were unsuccessful in their attempts to rule single-handedly. Today, we understand the term “Romans” to refer to the group of people living under Roman rule from the beginning of the Roman republic through the time described above, in which the leadership of the Roman Empire had been split, but both parts of the empire were still actively part of the Roman Empire, through the mid-400s.
However, after several devastating attacks in the 400s, the Western Roman Empire, with its main political base in the city of Rome, deteriorated quickly and succumbed to barbarian invasions, after which, most leadership and government in the Western Roman Empire broke up into small, tribal leadership based around barbarian tribes in Western Europe. The Roman Empire shrank significantly with these changes, but the city of Constantinople remained the political center of the Eastern Roman Empire, which is also called the Byzantine Empire, until its fall to the Ottoman Turks in 1453. Cultural and religious traditions grew and changed in the Eastern Roman Empire much differently than in the kingdoms of Western Europe. The term, “Byzantines” is best used to refer to the group of people who lived in the Eastern Roman Empire under the rule of the emperor in Constantinople between 476 and 1453.

The Byzantines were Romans, and that's precisely how they saw themselves. We often speak of the Roman Empire as one single entity. Yet for several centuries, it had been divided up into two parts, each with its own emperor: one in the West, centered on Rome; and the other in the East, whose capital was Constantinople. The vast geographical distance that separated the two halves of the empire inevitably led to the development of significant cultural differences between them. For instance, Romans in the West spoke Latin, whereas those in the East were mainly Greek-speaking. Nevertheless, inhabitants of both East and West still regarded themselves as partaking of a common heritage.
Although the Western half of the Roman Empire fell to the barbarians—actually Romanized Christians—the Eastern half survived as the Byzantine Empire, and would live on for almost one thousand years before Constantinople itself fell to the Ottoman Empire in 1453.