Naive set theory is the original set theory developed by mathematicians at the end of the 19th century, treating sets simply as collections of things.
Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as
Russell's paradox) in naive set theory. It treats sets as "whatever satisfies the
axioms", and the notion of collections of things serves only as motivation for the axioms.