Hi Aimee san; 下手 is the opposite from 上手 which means with good skills. While 私は料理が苦手です implies staying away from cooking from the first place, rather than questioning whether i am a good cook or not（probablｅly　not）. So, 苦手 can be used for stating tastes for foods movies　and people. 私は　ホラー映画が　苦手です。 does not mean 'not good', but 'do not like　horror movies'. hope it helps.

上手[うま(い)・じょうず]⇔下手[へた]。得意[とくい]⇔苦手[にがて]

Actually these are almost opposite... 上手[じょうず・うま(い)・うわて=good (in~), better (at~)...] vs. 苦手[にがて=poor (at~), weak (on~)]