下手 simply means "unskilled" and variants thereof. It is the antithesis to 上手, which can be seen in the kanji opposites.
苦手 on the other hand, while it can also mean "not good at", it contains more feeling you could say. Note that the first kanji actually means "suffering". So, 苦手 can be used for things you're not good at, or maybe things that you are slow to develop skill in because you simply don't like it. Maybe you're a swimmer, but you hate doing the butterfly. You're probably 苦手 then with the butterfly (not so much because you're not good at it, but because it's, for lack of a better term, emotionally painful to do it. 苦手 can even be used for food. A waiter might ask if there is anything you are 苦手 about. Perhaps you don't like raw onions. You are 苦手 with raw onions.
So, 苦手 can mean all of 1) poor/weak at doing something, 2) having a dislike of something, and 3) a recognition that something is hard to deal with.
I hope that helps!