I can think of one context in which it is always better to use "believe" over "think." When you want to convey your belief or trust in someone or something. God, for example. Being faithful, too. That your child will complete college.
In other contexts, I sometimes struggle to know which of the two words is appropriate. I understand that thinking is weaker than believing, but is the difference between them as simple as the one between thinking and strongly thinking.
In the following sentences, I'm not sure whether believe or think is the better choice.
- I believe/think you will win the competition if you practice every day for the next three months.
- I believe/think I got a B on the exam, but I'll have to double check. It's been a while since I saw my grade.
- I know that context always matters for questions like this one, but I believe/think you will be able to provide some insight anyway.