놓다 and 두다 both have very many usages, with significant overlap as well as distinctive ones.
Here are their most basic meanings.
- 놓다
1) release, let go (of something one has been holding). 잡고 있던 손을 놓았다.
2) put or place (at a particular location). 시계를 탁자 위에 놓았다.
3) put or render something to a certain state (when used as an auxiliary). 방을 예쁘게 꾸며 놓았다.
- 두다
1) put, place, or leave . 시계를 탁자 위에 두었다. 아이를 (남겨)두고 외국으로 떠났다.
2) keep for future use. 이 돈은 두었다가 집 사는 데 쓸 거다 (I'll stash this sum to use for buying a house)
3) have in certain way (direction, juncture, etc) 승리를 눈앞에 두다 (have victory before one).
Both are also commonly used with other verbs to modify the meaning (auxiliary verb). While they're often interchangeable in this usage too, 놓다 tends to be more about the current or near future outcome, whereas 두다 has a keener sense of doing something for future use.
- 다음 학기 대학 수업료를 준비해 놓았다 - I got the money together needed next semester's tuition.
- 다가올 동생의 대학 수업료를 마련해 두었다 - I saved the money for my brother's future college tuition.
- 일을 망쳐 놓았다 - (They) ruined the work. (놓다 can be used with unexpected outcome, like 버리다)
- 일을 다 망쳐 두었다 (X) - (두었다 can't be used for unexpected outcome)