아깝다 means
1. unwilling to spend (something)
ex)
그녀는 먹는 데 쓰는 돈을 아까워했다
She begrudged the money she spent on food.
이 신발은 버리기 아깝다
These shoes are still too good to throw away.
2. regrettable, pitiful
ex)
경치가 너무 아름다워서 혼자 보기가 아까웠다
The scenery was so beautiful that it was a waste to be seeing it alone.
그렇게 어린 나이에 세상을 뜨다니 너무 아깝다
It's such a shame that he died so young.
아쉽다. means
1. sorry, sad
ex)
이렇게 헤어지려니 너무 아쉽네요
It's too bad we have to part like this.
그가 없어서 못내 아쉬웠다
I felt something great was missing because he wasn't there.
2. feel the lack of (something)
ex)
그 당시에는 돈이 아쉬웠다.
Money was short at that time.
그는 아쉬울 때만 나를 찾는다
He looks for me only when he needs something.