According to the National Institute of Korean Language's etymology page,

날개 comes from -+-개, which is 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 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 래 came from 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).

According to the National Institute of Korean Language's etymology page,

날개 comes from -+-개, which is 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 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 래 came from 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).