"I think that ... not ... " and "I don't think that ..."

I think some differences may be pointed out, but usually it's just a matter of style and they shouldn't be any different.

Consider the following:

A strong atheist might claim:

I think there is no God.

Such statement naturally means that the claimer had, hopefully, gone through a long process of thinking before making this statement and that he really thinks there is no God.

However, an apatheist, a person who doesn't care about God's existence (doesn't think about the problem at all) might claim:

I don't think there's a God.

Which ultimately, if you're being shallow, might result in the same thing, but there might be a slight difference anyway.