Let me start with an illustration of these two verbs:
"I believe in the Gospel"
"I trust His promises"
After a simple Socratic Method I did on LigonierConnect, and reading Peter Enns' Why I Don’t Believe in God Anymore, I come to the conclusion that (yes semantics is important because we are human beings who communicate with words):
Believe: Relates to truth. Any representation of truth.
Trust: Relates to action.
Trusting a person means I am about to do something very specific according to that person.
Believing a person however, is more general if not specified. It can be an easy cop out. "Yes I believe you...good bye." Instead of "Yes, I trust you with my life, tell me what to do."
So, ironically, I have to agree with Enns:
I see the Bible focusing a lot more on something far more demanding: trust.
However, I do not agree with the way Enns discard the word "believe". Believe can be used as a foundation. Thus, "Believing in the Gospel" is necessary as a foundation. "Trusting the Gospel" sounds vague, but "Believe" here would cut it. Now, "Believe in His Promises" sounds too complicated, as if we are making room for many interpretations of these "Promises" of His. "Trust" is more simplistic and hands-on.
So each word has its own emphasis and use. To simply discard this as semantics is rather irresponsible, as the commenter (Terry) did at the Ligonier course I participated in. This is also something the charismatics often overlook. They conclude that they already have the right heart and any scholastic investigation that challenges their heart or their words is an affront to God. Hmmm...Come to think of it, that sounds like some extremists from certain other religion.