I think it does make a difference; I am telling T2 stuff about binge eating that I never would've told T1 because I wouldn't want him to think of me as some out of control overeater.
But the other side is that I felt more able to talk about other stuff with T1, like lack of self worth. I don't know why but I feel he wouldn't understand that like T1 did.
I think disclosure is always going to be difficult for people who are used to suppressing their feelings. Different T might shift that issue but not necessarily solve it.