Admittedly, this is inspired by EU4, but...
I think there's a tendency at times to assume that a Byzantine state would have a lot of the same goals as an Ottoman one. To an extent that will be true; both will want to dominate the Aegean, Balkan, and Anatolia. But beyond that?
I can't see the Byzantines expanding into Iraq or Egypt after the 12th century; the state never showed much interest in conquering large Muslim populations.
On the other hand, the Byzantines would probably exhibit more interest in conquering Italy than the Ottomans did. Perhaps they would also be more inclined to conquer the Crimea outright, and invade Ukraine?
Thoughts?