a. I hope you win next time.
b. I hope you'll win next time.
Are they both correct?
If so, are there any differences?
Yes, they are both correct. I would find it much more natural not to use "will" in that particular context. So I prefer (a), but both are correct.
There are no differences between them in that case. But take a case like this: "I hope he is there." Without context, "is" can refer to the present or the future.
There is no way, however, to suppose that "will be" in the sentence "I hope he will be there" can refer to the present. Thus, "will" can remove ambiguity sometimes.