To believe is to think that something is true. To trust is to put your confidence in another person that you have known for a period of time, like a good friend. Examples: He believes in Jesus. She believes BMW is a better car than Toyota. They believe that aliens do exist. I trust you will cal me if you have any problems. Eric trusted John to be his house sitter for 2 weeks. Can I trust you to keep a secret?

E.g: I don't believe the promises that politicians make and she's never lied to me before.

The word “believe” means that we believe someone’s statement, but the word “trust” means that we believe a person is honest and reliable in general.