For the most part, they are the same thing. But trust is something you've built up over time. You believe your friend won't steal your money. You trust your best friend won't steal your money. Still a belief, you've just had it awhile. Faith is another way of describing this concept.

There can be differences, depending on the context. Here is one difference, in many contexts: 1. You “believe” something because you have good proof. Maybe you have seen it with your own eyes. Maybe you have done experiments, like a scientist. Maybe you have thought long and hard about the subject, and formed a strong opinion. 2. “Trust,” in contrast, often implies that you are relying on SOMEONE ELSE to tell you what is correct. “I trust her financial judgment, and so I am going to invest my money the way she recommends.”

“I trust her” (anything she says I will believe) “I believe her” (I believe the specific thing she said, but I might not believe something she says later)