From my own research (and with the help of my friend that lives in Korea), here's what I've gathered:
- The Naver dictionary lists them as synonyms, and in most cases they can be used interchangeably.
한국어is of Chinese origins. It's 한자 (Hanja) is
語, meaning 'language'. The
한국말is native Korean (고유어).
한국어means a Korean language in general, but
한국말refers specifically to Korean speech. So you'd use
한국어when saying 'I study Korean', but
한국말when saying 'I speak Korean'. 말 can mean word/language/speech, but also 'talk', so this makes sense.
- 한국어 is generally more commonly-used
“한국어” and “한국말” are exchangeable. You can use both of them. The difference is “한국어” is mostly used in sentence or essay or in any formal sentence.
E.g. 저는 한국어 공부한지 1년 넘었어요 means “I have learnt Korean language more than a day already”.
On the other hand, “한국말” is mostly used in speaking. When you speak with Korean people, they usually use “한국말” and rarely say “한국어”.
E.g: 민아씨가 한국말 잘하시네요 means “Mina, your Korean is so good”.