不肯 emphasizes deliberate unwillingness, not mere inability. Compare: 不能 (cannot, due to ability/circumstances), 不会 (cannot, lack of skill), 不肯 (won't, by choice).
grammar
不肯 is always followed by a verb: 不肯说 (won't say), 不肯走 (won't leave), 不肯帮忙 (won't help).
