"He didn't come" is a simple fact about what happened: he did not come.
"He wouldn't come" suggests that he made a deliberate choice not to come. It means something like "he refused to come" or "he was unwilling to come."
"Why didn't he come?" asks for the reason why he did not come. "Why wouldn't he come?" asks for the reason why he was unwilling to come.
(Depending on the intonation and context, "Why wouldn't he come?" could also be a rhetorical question, suggesting that he obviously /would/ come. The sentence with "didn't" can't be used this way.)