There are some effects caused by the survival of ERE, no matter what her borders are, given it is more than a city-state:
1. Science/culture: 13th and 14th c. are known as the era of "Second Byzantine Humanism" and Palaeologan Rennaisance". The Byzantines, having access in a quite huge stack of ancient Greek works and a tradition in studying them, produced a cultural and scientific movement close to the equivalent "Humanism" and Renaissance" in Italy, only almost one century earlier. Note that the boom in the Italian Humanism came after the Florence Counsil, when the Byzantines sent a large delegate and its members were contacted by Italian scholars, and sold to them hundeds of books in order to cover their expences. So, a surviving Byzantium would have a huge effect to the progress of philosophy, art and science in Europe. Not imposible to see something like Enlightenment in the 17th c.
2. Religion: If the survival of Byzantine Empire does not come from a subversion to the Pope, the effect of the Byzies still out there would be significant for Greek Orthodoxy and European Christendom in general. First of all there would practically be no muslims in Europe (except from some in southern Russia, but who will have more or les the fate of OTL). Second, the Pope will still have a conciderable religious rival, and in order to face him, he has to cooperate more with some royalties, which means that he is less powerful than OTL. Furthermore, if something like Reformation happens, imagine the effects of a Greek Orthodox Empire do be a player in the chessboard: IOTL the Ecumenical Patriarch Cyrilos Loucaris had an intence corespondance with the Lutherans and become quite close to some of them, before he backed off due to the Sultan's dissaproval. Imagine A byzantine Empire and her Patriarch present in the Reformation: what would happen especially to Croatia, Hungary, Poland, Bohemia?
3. Geopolitics: with a surviving Byzantium there would be no Ottomans in the Balkans. That means that Hungary stays powerful, so that Austria either doesn't expand, or she expands westwards, creating a lot of problems within the HRE. Also, a surviving Byzantium means much less powerful Italian Merchant Republics, so some power like France, Spain or Austria, can conquer Northern Italy. Furthermore, except if the westerners insist on attacking Byzantium (which I doubt it can exceed early 14th c.), Spain does not face a great enemy like the Ottomans in the eastern Mediterranean, so she focuses on Maghreb, and probably builds a small empire there, includind northern Morocco, a part of coastal Algeria and northern Tunesia, by mid-16th c. In general there would be a faster expansion of western power in the East, which can be concluded in mid-19th c. instead early OTL 20th; if Northern Egypt (Nile Delta-Damientia/Suez stripe) is to be placed under western or Byzantine control early, there would be huge effects concerning european power expansion to Asia. Also, Russia has no Ottoman rival, so she expands easier and faster: she has some extra power to move east faster, or move west...
4. Economy: No powerful Italian Merchant Republics means a different history of european capital formation and movement. this could result either to a faster development of the Low Countries Capitalism and Industrialisation maybe a hundred years before OTL. In that case I can see Byzantium following this development to some extend at least, so industrialisation moves eastwards earlier (OTL was rather late 19th c., while it never had a core in SE Europe). A surviving Byzantium would also have effects concerning commerce with the East with the importance of Around Africa Routes lessened a lot: western merchant expansion in the eastern Mediterranean would be faster, so Portugal either performs worse than IOTL, or focuses more in the Americas. Probably a different kind of colonial expansion in Asia and Africa.