Well, with regard to the Roman Empire and Byzantine Empire, you need to understand their distinguishable religious, cultural and political characteristics.
The Byzantine Empire, during its first 300 years-(from Emperor Constantine until the rise of Emperor Heraclius), was essentially, an outgrowth of the old Roman Empire; that is to say, the Byzantine Empire was the Eastern half or zone of the Roman Empire, while Rome-(as well as the city of Milan beginning around 300 AD/ CE), represented the Western half or zone of the old Roman Empire. However, by 476 AD/CE, the Western half of the Roman Empire collapsed and were overrun by the Visigoths-(as well as related Germanic tribes), while the Eastern Roman or Byzantine Empire continued for several centuries-(until its collapse in 1453).
For the first 300 years, the Byzantine Empire was dominated and governed mostly by ethnic Roman Emperors-(i.e. Constantine, Justinian), though by 606 AD/CE, with the rise of Emperor Heraclius, the Byzantine Empire became an increasingly Hellenic and religiously Eastern Empire. This would last until "The Fall Of Constantinople" in 1453-(though the city of Constantinople was occupied by the Papal backed Crusaders during the first half of the 1200's).
While Greco-Byzantium flourished for much of the Early Medieval period, Rome, the majority of the Italian peninsula and nearly all of its Western and Northern European territories slumbered through "The Dark Ages". Politically speaking, many European territories were either under direct or peripheral control of the Papacy. Even the so-called, "Holy Roman Empire"-(in its early years), was still largely under Papal influence and orchestration. And "Dark Ages" Europe, was a Roman rite Christian religious culture largely influenced and controlled by the Papacy thereby distinguishing itself from the Byzantine Christian East.
So as you can see, there were major ethno-demographic, cultural, political and religious "distinctions" which contrasted Rome from Constantinople throughout much of The Middle Ages.