"아쉬움 and 안타까움" As I see it, the two are similar sentiments(sorry, sad) but differ primarily in the sharpness and nature of it. 아쉽다 means something leaves regrets in one's mind usually because it was "not enough" (서운하다 is similar). In a different usage, it can also mean that one is in need of something (돈이 아쉽다: I need money). 안타깝다 is typically a much sharper sorry feeling for something that is "not going as desired". So 아쉽다 is much like "regret" while 안타깝다 is closer to anxiousness or pity. For example, we might say 벌써 헤어지게 돼서 아쉽다 (I feel sorry we have to part so soon) but not 안타깝다. On the other hand, 시험에 두 번이나 떨어져서 안타깝다 is natural but 아쉽다 would sound too weak. An easy way to remember them is their spelling. 아쉽다 sounds smooth and has "쉽다"(easy) in it. 안타깝다 has 타 and 깝 which are much stronger sounds. It is probably not a random coincidence that it does.

'아쉽다'와 '안타깝다'는 결국 뜻대로 되지 않음에 대한 '탄식'의 의미를 내포하고 있다는 공통점이 있지만 기본적으로 '아쉽다'의 '탄식'은 화자(말하는 이) 자신의 일에 대한 탄식이며 '안타깝다'의 '탄식'은 화자가 아닌 상대방 또는 제 3자에 대한 '탄식'입니다. 아쉽다 means "feel regret for the matter of the speaker oneself" 안타깝다 means "feel regret for the matter of the listener or a third person" Thus, 아쉬움 is used for the matter of the speaker oneself 안타까움 is used for the matter of the listener or a third person