One of my Japanese dictionaries says: 下手[へた] ①（例）下手な説明。（意味）うまくできないこと。また、その人。(unskilled. someone unskilled. e.g. poor explanation) ② --省略-- 苦手[にがて] ① --省略-- ②（例）苦手の国語が好きになった。（意味）上手でないこと。自信がないこと。(unskilled. not confident. ) -------------------------- 広辞苑 says: へた[下手] ①物事に巧みでないこと。まずいこと。不器用なこと。また、そういう人。（例）「下手な歌」(unskilled in doing things. poor. awkward at~~. someone unskilled. ) ② --省略-- にがて[苦手] ① --省略-- ② --省略-- ③不得手[ふえて]。不得意[ふとくい]。（例）「数学は苦手だ」 (weak in~. not good at~. e.g. I'm weak in math.) ---------------- Hmm I think 下手 is more "objective" and 苦手 is more "subjective"; 私は料理が下手だ sounds like you're really a poor cook, and that's an obvious fact, while 私は料理が苦手だ sounds like you might be a poor cook but it might be just what you think, it might be you just don't like cooking, which is similar in meaning to 私は料理に自信がない=I don't have confidence in my cooking/私は料理が嫌いだ/私は料理が好きじゃない=I don't like cooking, etc. We also use 苦手 for school subjects you're weak in: 社会は得意ですが、理科は苦手です。(You don't say 理科は下手です。) for foods you don't like or you can't eat: 私はタコは好きなんですが、イカは苦手なんです。(You don't say イカは下手です。) for people you don't really like: 山田さんは苦手だ。(You don't say 山田さんは下手だ。) or for your weak point: ボクはシャイで、女性と話すのが苦手なんです。(女性と話すのが下手なんです would be fine, but 苦手 is more commonly used, and the 苦手 here implies that you're not very confident in yourself.) I think 下手 is more used to refer to someone (not yourself) unskilled/poor at/weak in~, than 苦手: I'd rather say 鈴木さんは運転が下手です(Mr Suzuki is not a good driver) than 鈴木さんは運転が苦手です (and I think this also is partly because 下手 is more objective and 苦手 is more subjective.)

"下手" by Kanji means "lower hand" and is reffered to those who are lower than the speaker. "苦手" by Kanji means "bitter hand" and is reffered to those who cannot complete a given task. A bad worker in a sense. Written out, they are different, but are commonly spoken the same way.