I think you need to assume that $V$ and $W$ are smooth varieties; then you'll find the statement (or, at least, a very similar one about stalks) in SGA 1, II, Corollaire 4.6 or (in English) Bosch, Lütkebohmert, Raynaud, Néron Models, 2.2, Proposition 10.
I've been trying to come up with a good counterexample with non-smooth varieties. Perhaps you can do something like this: take $W = l_1 \cup l_2$ to be the union of two lines in the plane, and $V = l'_1 \cup l'_2 \cup l'_3$ the union of three lines in the plane meeting in a point. They both have one singular point, at which the tangent space has dimension 2. Map $V \to W$ by sending $l'_1$ to $l_1$, and both $l'_2$ and $l'_3$ to $l_2$, so that the singular point of $V$ maps to the singular point of $W$. Then I believe you get an isomorphism of tangent spaces at each point, but the map isn't étale.