Trust would actually mean that a person places complete confidence in another person and can rely on them for anything. Believe is a more temporary concept that requires the person to place faith in a person for a select time frame. Believe can also refer to accept that something

You use either one to say you think what someone has said is true (i.e., I trust what you say/I believe what you say), but they are not always interchangeable. At least in North America, we generally only use "believe" to state a hypothesis or idea (i.e., I believe that English is useful). We use "trust" in at least two other ways. One is to say that you consistently have faith in someone or something (i.e., I trust my friends). The other is to express a hope (i.e., I trust that you will make the right decision). That said, I should also mention that I personally find this last use of "trust" very formal and even kind of snobby and condescending. But maybe this is just the variety of English I speak or my personality.