"God exist" Vs. "I believe God exist" Vs. "I believe in God"
Do you see a difference between these three statements and if so what are those differences?
They're essentially all saying the same thing. The differences are generally contextual and rhetorical.
"I believe god exists" and "I believe in god," are basically the same statement, and their usage could often be explained simply by usage trends, but I suspect that there is probably, more generally, an implication of trust (faith) when someone says "I believe in god."
"God exists," is probably most often used as a rhetorical tool to try to limit and control the conversation by expressing a belief as an absolute fact. This, of course, only works with people who already share your belief, and comes off as silly and obstinate to people who don't.