One open conjecture is that every even integer greater than two is the difference of two primes. (Some superficial discussion here.)
Goldbach's conjecture states that every even integer greater than two is the sum of two primes.
The big question: are the two equivalent? That is to say, do these conjectures imply each other? I spent a bit of time pursuing this question, and I did not find a satisfactory answer.
I now suspect that the two are actually not equivalent -- if they were, then I think it would suggest a symmetry on the prime numbers that I don't think they have.
Anyway, I'd be glad to hear your input on the matter.