Well, even with a Muslim West and a Christian East you'd still have the geographical features associated with Europe and the Middle East: Europe being far away from Asia, lots of navigable rivers and plains to create capital-intensive regions, plenty of opportunity for naval development and easily fragmented into separate states by mountains and other obstacles, etc.
Meanwhile, the Middle East will continue to be an important conduit from China to Europe, but would still suffer from rugged terrain, nomadic raiders, and relatively poor land which requires some degree of capital investment (qanats and the like) to develop properly.
So the geographical imperatives for Muslims to go to find China via sea routes, collapse into competing nation-states, and maybe even develop forms of capitalism still exist; while the imperatives for the Middle Eastern Christians to find alternate naval routes for China would be much reduced.
Ultimately the question boils down to 'what is different between Islam and Christianity in practice', which is kind of a risky topic to be discussing in any case.
My two cents: I don't think Islam in Europe can fuel a scientific revolution, considering that Islam at its core (Koran, hadith) arguably inserts itself into so many more facets of life; and in any case, neither Catholics nor Mainstream Protestants argue that the Bible is the literal word of God, unlike the Koran. All this gives Christianity much more leeway for logical insinuation/rationalization (what would be ijtihad in Islam). An al-Ghazali - somebody who essentially rejected Greek philosophy and logic as being dangerously impious - would have much harder time formulating a rational argument in the Christian world (for one, he can't argue that the Bible gives 'absolute certainty', unlike philosophy).
However, one could argue that if the Muslims entrench themselves in Europe and imbibe the Greek/Roman culture more deeply, their theology would also become more 'Greek-related' and therefore be more open with regards to these aspects of their religion. This is, of course, not to say that Muslims can't be prosperous, colonize or even industrialize in Europe... I just don't see them making the scientific advances that could make their progress approach OTL levels.
As for the Middle Eastern Christians: assuming the Mongols (or some other horde) still come down and wreck everything it's entirely possible for the more fundamentalist forms of Christianity to gain in popularity as a response to this sort of existential crisis. Stagnation seems eminently possible considering that it won't be Christians who discover America in this scenario nor will they likely participate in colonial efforts. Christian-Muslim war would probably become much more likely considering that Christians would own Mecca/Medina and the Muslims would probably want that back.
So in conclusion, a Muslim Europe vs. a Christian Middle East will probably lead to a world that is comparatively less developed than OTL, at least in the scientific field. Europe would still probably be the continent which colonizes the Americas, starts the modern Industrial Revolution, and eventually dominates the world. Of course a lot of what happens would have to be determined by how the Muslims got to Europe and vice versa, but generally speaking I think the outcome would be something like this.