Quoted from http://blogs.koreanclass101.com/blog/2008/08/06/whats-the-difference-between-%ED%95%9C%EA%B5%AD%EC%96%B4-and-%ED%95%9C%EA%B5%AD%EB%A7%90/
Both the words ‘한국어’ and ‘한국말’ are composed of two meaning blocks : Korean + language.
어, which can be written in the Chinese character 語, means “language”, and 말, which is a native Korean word, also means “language”. So basically 한국어 and 한국말 are the same meaning, but I’d say 한국어 is a little more formal word while 한국말 is a more colloquial word.
‘어’ can be found in many other language names, but ‘말’ is not so commonly used other than in the word ‘한국말’ or ‘우리말(uri mal - ‘our language’, which is a word that Korean people use to mean ‘Korean’).
Here are some examples:
English = 영어
Chinese = 중국어
Japanese = 일본어
Arabic = 아랍어
Spanish = 스페인어
French = 프랑스어
Latin = 라틴어
Russian = 러시아어
‘말’, however, is a native Korean word, so it’s used much more commonly in everyday conversations. And in this case,말 also has the meaning of “words” and “what one says” as well as “language”.
For example, (the following sentences are all in the intimate politeness level)
내 말 안 들려? = Can’t you hear me?
말 조심해. = Watch your language.
내 말 무시하지 마. = Don’t ignore what I’m saying.
말이 씨(seed)가 되는 거야. = Your careless words might turn into reality.
그런 말 하지마. = Don’t say such a thing.
무슨 말이야? = What do you mean?
I hope those sample sentences are helpful, and in conclusion, 한국어 and 한국말 are the same thing, but 어 is a dependent Chinese-character-based letter to mean ‘language’, and 말 is a word of a wider variety of meanings. ^^