To believe and to think mean the same thing when you are referring to the action of expecting or supposing or to have an opinion. 'I believe/think they will arive soon'. But both verbs have other meanings which are not synonymous.
Believe can mean a) to accept as true or real 'Do you believe what he said?' b) to have faith confidence or trust. 'I believe in God.' c) to have confidence in the value of something. 'I believe in democracy.'
Think can mean many different things including a) to judge or regard ' I think it is unfair.' b )to intend. 'They thought they would take the train.' c) To remember. 'I think her name is Julia.' d) To have care or consideration. 'Think of others before yourself.' e) to imagine. 'Think what he will say when he sees you.' f) to consider an idea. 'They are thinking of buying a new house.' g) to decide by reasoning or reflection. 'He's thinking what to do.'