下手 and 苦手: weak skills and uncomfortable situations
Distinguish poor skill from discomfort or aversion, including 苦手 with foods and people.
Lack of skill
私は絵を描くのが下手です。 says that the speaker is poor at drawing. 下手, read へた, evaluates an action or result as unskilled.
字が下手なので、ゆっくり書きました。 says that the speaker wrote slowly because their handwriting is poor. The slow writing occurred, but its quality afterward is not specified.
An activity that causes difficulty or discomfort
人前で話すのが苦手です。 says that public speaking is difficult or uncomfortable for the speaker. 苦手, read にがて, can involve emotional discomfort as well as ability.
泳ぐのは苦手ですが、練習しています。 says that the speaker is not good at swimming but is practicing. The current weakness need not be permanent.
Difficulty with foods or people
辛い食べ物は苦手です。 says that the speaker struggles with or dislikes spicy food. It does not evaluate cooking skill and cannot simply be replaced with 下手.
あの人は少し苦手です。 says that the speaker finds that person somewhat difficult or uncomfortable to deal with. It does not judge the other person's skill; comments about people require attention to context.
Does 辛い食べ物は苦手です mean being poor at cooking spicy dishes?
Show the answer and why
It normally means having difficulty eating or not liking spicy food. Cooking ability would need the cooking action specified.