[hoping1]

You probably mean set theory instead of graph theory, since set theory and category theory are kind of seen as two foundations for math.

Both category theory and set theory use sets. But set theory tries to make absolutely everything into a set. It takes on a little complexity in this quest, because of Russel's paradox. Category theory can be seen as studying set theory, among other things, so it is "bigger" or more all-encompassing than set theory. Many things studied in category theory aren't possibly sets.

Set theory is more immediately intuitive, but category theory organizes things in a way that are ultimately more insightful, I think. Meaning that once you can get into the category theory headspace and learn to navigate it, it becomes a much better environment for thinking without mistakes. In my personal opinion.

[Koshkin]

Also, morphisms from A to B form a set.

[hoping1]

Only in locally-small categories. But yes, category theory often makes use of sets.