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.
