It seems weird to me that a lot of entities used the title for a long time after Rome fell (Kaiser, Czar), but the ones that were most closely related to the roman empire didn't.
For a while, Caesar was used to designate the appointed heir of the reigning emperor, and this became even more evident during the Tetrarchy, with two Augustus and two Caesars, one of each for the eastern and western administrative parts of the empire.
However, the title lost its "imperial importance" when the emperors in Constantinople began to elevate their chosen heirs as co-emperors (Symbasileus). In turn, the title of Caesar began to be awarded to other close relatives, such as younger sons. Then Alexios I Komnenos creates the title of Sebastokrator (a mixture of sebastos and autokrator), making the title of Caesar lower in the hierarchy. By the time of Manuel I Komnenos, the title of Caesar was further diminished, when the honorific of Despot appeared, which was used for a time to designate the heir to the throne (just like Caesar was once used).