Differences between pure and impure set theory? 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?
 A: One big difference is that models of impure set theories admit nontrivial automorphisms. Consider $\text{ZFU}$ for instance. Take a set $U$ of urelements and a permutation $\pi : U \to U$ of these urelements. Then $\pi$ extends to an automorphism of the (impure) set theoretic universe by
$$\pi(x) = \{ \pi(y)\, :\, y \in x \}$$
for all sets $x$. This is a well-defined recursive definition because by the axiom of foundation you'll always eventually hit the empty set or an urelement.
This kind of construction allows you form models of $\text{ZFU}+(\neg \text{AC})$. Take a group $G$ of automorphisms of $U$, which extend to automorphisms of the universe as above. Then under a suitable definition of 'symmetric', the hereditarily symmetric sets form a model of $\text{ZFU}$ in which the axiom of choice fails.
This is impossible in $\text{ZF}$ because in this setting the only automorphism of the universe is trivial. In fact, even if you admit choice in the construction given above, even though choice fails in the symmetric submodel, it still holds in the class of pure sets of the submodel.

Further reading: The Axiom of Choice by T.J. Jech.
A: 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. 
