Isn't "believing that God doesn't exist" the same thing as "not believing in God"?

No, it's not.

Let's break down the "equation":

Believe no God = No belief in God

(Yes) Believe No God = (No) Belief God

(Yes) Belief = (No) Belief

Yes = No

Obviously, it's illogical. One can't dismiss this as "purely semantics". It's actually indicating a subtle, yet critical issue.

The difference is the level of assertion.

To say that not believing in a god is the same as believe there are no gods, is like saying that not believing there's an odd number of stars in the universe is the same as believing there's an even number. Your position can be that both claims are undemonstrated.

Many atheists are "neutral", in that they are not necessarily accepting that a god absolutely does not exist, but rather are waiting for theists to demonstrate their claims, using the scientific method, preferably.

Colloquially, most people think of this position as "agnostic", but in fact, it's both. In short, agnosticism deals with what is knowable, and atheism deals with what's believed. Most self-described atheists describe themselves as "agnostic atheist".