It's a complicated question! but in simple terms.. to hope = to be optimistic that something will happen, and maybe it will. I hope they can come to the party (There is a chance they will come, I am optimistic that they will) to wish = to want something to happen, but it isn't going to happen or it is very likely to happen I wish they were able to come to the party. (I want them to come, but they aren't going to come because..)

Mike's answer above is correct, but I would like to add a little to it. There is also a difference in the grammatical construction. Hope usually takes a future tense, where as wish takes past tense. In Mike's example, "I hope they can come to the party" or "I hope they will come to the party" takes a present or future tense of the verbs -- can and will. Whereas, "I wish they were able to come to the party" or "I wish they could come to the party" takes a past tense version -- were and could.