부르다 is actually "to call out", but in counjunction to "song", it's turned to "to sing" 노래를 부르다, or 노래부르다 This expression is strictly related to sing songs vocally. 노래하다 is similar to above, but it isn't limited to vocal, and can also mean "to praise", "to admire" in any kind of (artistic) performance. 어머니는 내가 어릴 적 많은 고전민요를 불러주셨다. My mother used to sing many classic folklores for me when I was a child. 대자연의 아름다움을 노래하는 시를 쓰시오. Write a poem praising the beauty of mother nature.

You can find "부르다" in dictionary. "부르다" have much broader meaning. it can be used to making sound. on other hand, "노래하다" available only to sing a song