"I am not in trouble."
I think I'm definitely missing the same nuance as you, but my guess is that this is a case of, uhhh, I'll call it past-tense-continuance, like how "I am hungry" is literally "my stomach became empty". Maybe こまりません has an implication more like "I'm not getting in trouble (or becoming troubled?) right now."
Plugging it into Google Translate (with kanji: 困りません) gives: "I do not need help." Not sure if that supports my hypothesis or not, 'cause, y'know, Google Translate.
I think they both can be replaced with 大丈夫 (だいじょうぶ) です, but 困りません means more like "won't get in trouble" whereas 困っていません refers to the present situation.
傘を忘れても困りません。I (or you) won't get in trouble even if I forget my umbrella. 傘を忘れても困っていません。 I'm not in trouble even though I forgot my umbrella. / I haven't got in trouble even though I forgot my umbrella.