Does the nonexistence of urelements ever matter? As far as I'm aware, when doing mathematics (apart from when doing axiomatic set theory) the assumption that all objects are sets, and that there are no urelements, is never actually used. When reasoning about a group, or a topological space for instance, it never matters that the elements of the structure are themselves sets rather than urelements. Are there any general results to this effect - showing that if one works in ZA say (Zermelo set theory in which atoms/urelements may exist), one can establish the same mathematical results in some sense as if one works in Z (Zermelo set theory)? For instance if $T$ is a second order theory and $\phi$ a statement in the language of $T$ such that $\text{Z}\vdash(T\vDash \phi)$, does it follow that $\text{ZA}\vdash(T\vDash\phi)$?
 A: To some extent, but not quite there, you're right. Urelements (or "atoms") are kinda irrelevant. When you want to reason about structure, the underlying set is irrelevant, and so whether or not your set is pure or with urelements is not important.
Nevertheless, if you want to reason about the universe, urelements may play a role. In the study of the Axiom of Choice, for example, using urelements is an easy approach which is considered clearer than applying forcing-style arguments. And we even have abstract transfer theorems letting us move from urelements to results in ZF. But it is not true that everything can be transferred.
For example, "The power set of every ordinal can be well-ordered" implies the Axiom of Choice in ZF, but not in ZFA. If you want to learn more on these kind of statements, Jech's "The Axiom of Choice" has a nice chapter about this which covers the basics.
Other examples of this flavour come from a proper class of urelements (e.g. here and there on this very website) being used to separate other forms of Global Choice or Replacement-like axioms. (One thing to note is that we can replace Urelements by Quine atoms, i.e. sets satisfying the equation $x=\{x\}$, which is why we can think of pure sets as something satisfying the Axiom of Regularity/Foundation.)
