I believe there is not a god.
The placement of the word “not” before god, rather than believe, indicates that there is a belief against the existence of a god. In the same way that you might say “I believe my mother is not at the store.” You have strong evidence to support that she is at the doctor, possibly, which in turn serves as evidence that she is not at the store.
I do not believe there is a god.
By placing “not” before believe, instead of god, you indicate that there is an absence of belief, rather than a firm belief in the absence.