In Stacks 02GH it is mentioned étale morphisms are locally standard étale. This seems to be the analogue of the local form of local diffeomorphisms in the smooth category. In the latter, local structure plays a big role.
Where is the local structure of étale morphisms needed further on in the theory? Which important proofs crucially require actually knowing an explicit local form?
One part where it is essential is that the etale site is a "topological invariant". More precisely, if $X_0 \subset X$ is a subscheme with the same underlying space as $X$ (for example, some nonreduced structure), then there is an equivalence of sites: $Et/X \to Et/X_0$, given by $Y \mapsto Y_0 = Y \times_X X_0$.
The proof of full faithfullness doesn't require the local structure, rather it goes through some elementary facts about sections of the projection maps from the product and some more "Hartshorne-style" facts about maps. To show essential surjectivity though, you construct it locally. If you can show that for any point $p$ in the $X_0$-Scheme $Y_0$, there is a neighborhood $U_0$ so that $U_0 = U \times_X X_0$, then a standard glueing argument for this will complete the proof. However to show this local fact, you need (very crucially!) the local structure of an etale map. This is contained nicely in Lemma 2.3.11 in Lei Fu, Etale Cohomology Theory.