I think wintertriangles has it right. I have both an 이불 and a 담요 on my bed. Following are some sentence examples from Naver.
이불 --blanket, covers, (Am) comforter, (Brit) duvet; (침구류) bedding, bedclothes; (누비이블) quilt
Idiomatically, “wetting the bed" is 이불에 오줌을 싼다.
어머니는 아이에게 이불을 덮어 주었다 The mother covered her child with a blanket.
아이는 일어나자마자 이불을 갰다 The child put away his bedding as soon as he got up.
그는 잘 때 이불을 걷어차는 버릇이 있다 He has a habit of kicking off the covers in his sleep.
담요 潰― blanket
담요를 덮다 pull up a blanket
그는 담요로 몸을 감쌌다He rolled himself (up) in his blanket.
The Hanja for 담 is indicated incorrectly in the Naver English dictionary. It should be 毯 instead of 潰. 潰 is read 궤 and has nothing to do with blankets meaning-wise either (its basic meaning is "to break (of dams or similar)", cf. http://m.hanja.naver.com/search/?mflag=1&query=%E6%BD%B0).
담요 and 이불 are both blanket, but I think 이불 is a thicker one. 담요 might be a general blanket you'd use on a sofa or something.