TITLE: Differences between pure and impure set theory?
QUESTION [6 upvotes]: What are some differences between pure and impure set theory?
For example, this paper references the result that ZFC with urelements is categorical if you assume that the urelements form a set. ZFC, however, is not known to categorical.
Likewise, there are important metatheoretic differences between New Foundations (NF) and NF with urelements. For example, there is a relative consistency proof for NF with urelements, but none is known for NF.
Are there any other major differences like these between set theories and their impure counterparts?

REPLY [2 votes]: As well as Jech's excellent book, you might also be interested in looking at one of Michael Potter's two set theory books, particularly the later version Set Theory and Its Philosophy. Here he principally develops $\mathsf{ZU}$, a version of what has become known as Scott-Potter set theory with urelemente. Potter gives reasons too for why it is natural to start with, and continue to work with, theories with urelemente.