What is the difference between these two sentences?
It's not a sentence, there is no predicate. It's a noun phrase meaning "a person who likes fish", or if the general subject where this expression appeared is a comparison between different animal species (including human) "fish-liking human(kind)".
This on the other hand is a full sentence. "Human(kind) like fish".
I agree with macraf. They have a few differences.
This is a noun phrase because there is no predicate (述語).
This is a sentence because it ends with a predicate, na-adjective 好きだ.