According to the [National Institute of Korean Language's etymology page](http://korean.go.kr/front/wordLocal/wordHistoryList.do?mn_id=116),

날개 comes from [![][1]][1]-+-개, which is [![][1]][1]다(to fly, Modern 날다) + -개(suffix meaning "a tool to do such action").

Other words with -개 suffix are:

> 덮개, 지우개, 이쑤시개, 베개, 마개, 깔개, 끌개

나래 is just another form of 날개, which appeared as [![][2]][2]래 in the 15th century, which is unfortunately is as far as we can go back in the Korean Language. But linguists from NIKL think that [![][2]][2]래 came from ![][1]개 where ㄱ changed to ㅇ after ㄹ, which was a common phenomenon(sonorization of laryngeal consonants, Korean 후두유성음화) at the time. Note that ㅇ had a sound value [ɦ] back in the 15c.

The etymology page explains:

>아마도 동의어 ‘날개’와의 경쟁에서 밀려나 특정 지역에 국한되어 쓰이고 있는 것이 아닌가 한다.

> *It probably fell behind in the competition with the synonym '날개' and is now being only used in a specific area(in literature).*

  [1]: https://i.sstatic.net/0dFoS.png
  [2]: https://i.sstatic.net/hDt1C.png