The meaning of sentences using "wie" or "nach" in this context can be very similar:
Das klingt wie Gewehrschüsse!
Das klingt nach Gewehrschüssen!
(That sounds like gunshots!)
Differences I notice are:
The meaning seems to be the same, or almost the same.
There is a grammatical difference.
The second one is formally correct, but I am not sure whether the first is strictly correct too. It is certainly understandable, and also acceptable as a correct sentence for practical purposes - but it may not be theoretically correct.
There is some kind of subtle difference in style - but what is it? Maybe the first is more common in spoken language, and the second in written language?
These are all details - but it feels like I am missing something. Probably the difference on a more abstract level, like "The sentences are of different types, which are ..."
How can the sentences be described?
Do they mean the same?