There is very little difference between the two, and they can sometimes mean the exact same thing. However, sometimes "wish" is used for something that isn't possible, whereas "hope" is used for something you want to happen in the future. Examples: "I wish I could speak every language in the world" (In this case, it is an unrealistic hope that the speaker knows will not happen). "I wish I could come see you at Christmas time, but I have to be in California for a conference." (Another case where it is for something that is not possible, but stresses a desire for something else). "I hope you feel better soon!" (In this case, it is something the speaker wants to happen in the future) "I hope I do well on the test" Hope can also be used as an abstract idea, as in the expression, "my hopes and dreams." (Things that you want to happen in your future).