Good question, it's a bit tricky so I asked a Brazilian about this. The difference simply comes down to whether or not you are using the word "the" (the definite article).

• The verb gostar always requires the preposition de (of).
• In "gosto de música brasileira," you are just using de. It translates to "I like Brazilian music" in a broad, general sense.
• In "gosta da cultura brasileira," the word da is a contraction of de + a (of + the). So, you are literally saying "likes the Brazilian culture."

In Portuguese, certain nouns naturally attract the definite article more than others, but both sentences are grammatically correct. You could even say "gosto da música brasileira" if you wanted to be slightly more specific. But then it would sound more if you are saying that you like a specific Brazilian SONG (because "música" is a word both for "músic" (broad sense) and "song").