There is clearly a difference between “I don’t believe a god exists” and “I do believe that no god exists”. One is a positive statement of a negative while the other is a negative statement of a positive. The net result is the same.
It depends how the first statement is meant. But in general they are two rather distinct thoughts. The first statement is one an agnostic can make. The second one is not one an agnostic can make.
It’s quite possible that the lives of speakers who made each of these statements would show little difference. However the attitudes, actions and certainly discussions with theists might be radically different, given the potential epistemological differences between the two speakers.