Jisho

苦手 is slightly broader and thus vaguer than 下手. This makes it easier to use in more situations, and Japanese people like being a little vague. As you can see in the second definition, it can just mean it's not something you're fond of, rather than being bad at it. So unless they press further, people won't know for sure which you mean, and that's a more socially comfortable situation often.

下手 is very blunt and disparaging, so people probably won't use it when talking to someone in a polite context, even if referring to themselves.

An unambiguous, but less blunt, way to say you are bad at something is 不得意.