Let's put them both into a sentence and see if the meaning is the same or different.
"I think the Earth is flat." means that you have considered the idea of the Earth being flat, round, square, or held up on the backs of elephants, and in your opinion, the Earth is flat. In this scenario, there is a possibility that you might be willing to think about it again.
"I believe the Earth is flat." means that you don't need to consider the idea, you already know the answer that you agree with. In this scenario, you are less likely to be convinced that your opinion in not correct.