Precise relationship between elementary and Grothendieck toposes?
There are known statements that are true in any Grothendieck topos, but not in every elementary topos with NNO. For instance:
Freyd's theorem that a complete small category is a preorder is not constructively provable and can fail in elementary toposes with NNO, such as the effective topos; but can be shown to hold in any Grothendieck topos by an "external" argument (see this question).
Axioms that assert things like "the axiom of choice fails in at most a small (i.e. set-indexed rather than proper-class-indexed) way" often tend to be true in Grothendieck toposes (at least when defined over ZFC), but can fail in elementary toposes that correspond to things like "permutation models over proper classes". Some "small choice" axioms are WISC, AMC, SVC, and SCSA, but I don't know offhand the situation for all of these regarding their truth in Grothendieck topoi.
An axiom scheme that holds in all Grothendieck toposes, but not all elementary toposes with NNO, is "autology" (representability of large quantification in the stack semantics) as defined in this paper. This is a sort of "replacement axiom" for topos theory, and fails in models like $V_{\omega\cdot 2}$.
A Grothendieck topos (defined over ZFC) also has all higher inductive types, including in particular localizations at any set of maps, and free algebras for all (even infinitary) algebraic theories. These can also fail in elementary toposes with NNO, such as a model of ZF in which $\omega$ is the only infinite regular cardinal (see Blass's paper "Words, free algebras, and coequalizers").
Regarding the bullet point list of questions at the end, I believe that every true $\Pi^0_1$-sentence holds in the internal logic of any Grothendieck topos, so in that case $T_{\mathbf{GrTop}}$ is definitely not recursively axiomatizable.
If I understand correctly what you mean by "its contents depend on set-theoretic issues," this is already answered by some of the axioms listed in Mike Shulman's answer. In order to show WISC we need to assume the axiom of choice in the background set theory, and if we only work over $\mathbf{ZF}$ then e.g. $\mathbf{Set}$ is a Grothendieck topos where we can't prove that WISC holds.