Stylistically is the adverb of stylistic. It tells you the sentence is talking about the HOW of a piece of art or writing — its manner and methods — and not about its meaning, its message, or its quality.

You will most often meet it at the front of a sentence, doing a job I would call 'setting the frame': 'Stylistically, the two novels are very close.' The speaker is saying: let us put content aside for a moment and look only at the way it is written. English does this with a lot of -ally adverbs — historically, technically, politically — and stylistically is one of the family. Once you see that pattern, whole sentences open up.

It is also very common in the phrase 'stylistically different' or 'only stylistically', where the writer is gently saying the difference is on the surface. If two translations are stylistically different, they mean the same thing but do not feel the same.

Like stylistic, it lives in reviews, essays, and classrooms. It is a long word and a formal one, and a native speaker chatting about a song would just say 'it sounds different'. That is fine. Knowing a word does not mean you must use it — but you should be able to read it, because critics use it constantly.