"Show me just what Mohammed brought that was new, and there you will find things only evil and inhuman, such as his command to spread by the sword the faith he preached." - Manuel II Palelologus
There was a hereditary link too. From Wikipedia:
...The nephew of the last Emperor, Constantine XI, Andreas Palaeologos had inherited the defunct title of Byzantine Emperor and used it from 1465 until his death in 1503....At his death, the role of the emperor as a patron of Eastern Orthodoxy was claimed by Ivan III, Grand Duke of Muscovy. He had married Andreas’ sister, Sophia Paleologue, whose grandson, Ivan IV, would become the first Tsar of Russia (tsar, or czar, meaning caesar, is a term traditionally applied by Slavs to the Byzantine Emperors). Their successors supported the idea that Moscow was the proper heir to Rome and Constantinople. The idea of the Russian Empire as the new, Third Rome was kept alive until its demise with the Russian Revolution of 1917.