Well, there are several differences, and then there are the interpretations of these differences, and then there is what plays out in practise -- all of which are very, very different.
In the United States, we have the clause in the First Amendment to The Constitution which, in part, states: "Congress shall make no law respecting the establishment of religion." This, in most interpretations, hinges on the precise meaning of "establishment", and refers to the establishment of The Church of England as the "preferred religion" of the land. That was not going to fly in the nascent USA because of the disparate religious backgrounds of the constituency. Thus, the "Establishment Clause" was inserted as an afterthought to the original document. Now, in practise that's not actually the case. Even though there is no "Established" (i.e. lawful) religion, and that should be a protection -- it isn't. In the USA, it's evangelical Christianity that's wagging the State at the moment and calling most of the shots domestically (save for economically; that's another story).
In Europe, various countries still have "established religions", but which are largely ignored as being irrelevant in day-to-day life. Thus, there's not much problem, and things are the same as they've always been but "we've just moved on". Not the case in the USA; which has demonstrably gone backwards.
