I think it comes to to scientific evidence in regards to belief for some. Me, I personally don't know, nor will I ever know if there is some supernatural power looking over everything and everyone, although it's difficult to imagine.
As to your statement, god exists I think is a broad statement in which someone can take differently. To me, it's like a personal opinion that's being forced at everyone who reads it. It's like there are facts to prove that God exists, so it's not a belief, it's just god exists.
For the I believe god exists, that's merely someones belief. To me, it sounds less invasive and is simply stating their opinion. This could do with like religion, personal experiences, or cultural influences I think.