To me, a native speaker in the USA, they mean the same thing. I think that "to gain someone's trust" is more common here because it appears more often using a search engine than "to gain someone's confidence."

I think they mean the same thing. Analysing the word "confidence", it comes from "con-" (= "with") and "fido" (= "trust"). Someone that you "confide in" has your "confidence", which literally means that you "trust" them.