Both 'believe me' and 'trust me' are saying "Accept what I am saying as true" or "You can rely on me". The difference between the two is 'trust me' is more about accepting what I am saying because of my character only. You know my good character, therefore you know you can rely on me and trust me. In contrast, 'believe me' is more about accepting what I am saying soley because of my words and what I have told you. You don't know my character, but you can believe my words and the facts I have given you. Both expressions do involve accepting what I say as true, BECAUSE of my character and the words/evidence I present to you. But, with 'trust me' there is far more emphasis on my 'character' as the reason for accepting what I say.

"Believe me" in an acceptable thing to say, while "trust me" is a pompous and irritating phrase that only silly people use.

Thank you! I completely understand it.

Greg has given a good answer. In general, "believe" refers to believing statements of fact (you believe that something is true), but "trust" can refer to both statements and actions/behaviors. "Do you trust him?" can mean "Do you believe what he said?", but it also has a more general meaning of "Do you think he is a reliable/honest person?" If you don't believe someone, it means you think they are lying. If you don't trust someone, it could mean that you think they're lying, or it could mean that you think they might betray you, steal from you, or do something that would harm you. You can say "I don't trust him" even if you believe that everything he says is true. If you're telling someone to believe that what you're saying is true, then "believe me" and "trust me" have similar meanings (with a slight difference, as Greg has explained). But you can also use "trust me" in other situations, where it means "you can rely on me" or "don't worry, I'm not going to let you down."