한국어, 한국말 is same meaning, but the slight difference is that:
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 한국어 is a little more formal word while 한국말 is a more colloquial word.
1.저는 한국어를 공부합니다
2. When a Korean nativer speaker hear a foreigner speaking Korean and a Koren person says:
와, 한국말을 잘하시네요 (correct)
와, 한국어를 잘하시네요 (not natural)
Hope it can do help