There is, necessarily, some difference, but I don't think it's a matter of the brain functioning drastically differently.

I don't think so either. From the explanations others have given me, belief seems to come from a) wanting the world to be a certain way, b) thinking (probably not consciously) that the world is this way if they believe it is, and c) interpreting observations about the world as evidence for the world being that way. Well, I have a) as well. It would be really freaking awesome if there was a God who talked to you and answered your prayers and never let anything bad happen. But I know that my believing that doesn't make it true, and so I interpret the same real-world observations as meaning different things.