The difference lies in the means you have at your disposal to settle the membership question. Even within "Set Theory" (as an umbrella term), there are several

The difference lies in the means you have at your disposal to settle the membership question. Even within "Set Theory" (as an umbrella term), there are several axiomatizations of sets that define in different ways what a set is, how sets come into existence, what it means for an element to belong to a set, and what are the allowed ways to conclude that it does or does not. The link posted in a sibling comment by @empath75 goes into more detail.