The etale topology (ignoring etale cohomology) underlies the theory of algebraic spaces (which in turn is the basis for the "modern" approach to moduli spaces following Artin).
More concretely, the Artin approximation theorem provides a genuine sense in which the etale topology is the algebraic version of working locally for the complex-analytic topology (similar to the way that the implicit function theorem ensures the existence of convergent solutions to systems of analytic equations admitting a formal solution): in the stronger form proved by Popescu (building on Artin's results) it says that a finite system of polynomial equations over an (excellent) local noetherian ring with a solution in the completion admits solutions in a local-etale extension (even arbitrarily close to the solution in the completion).
Application: the Artin-Popescu approximation theorem enables one to "promote" solutions to relative algebro-geometric problems over completions (built via deformation theory, for example) into solutions over an etale neighborhood of a point in the base. More concretely, it implies that if two algebraic varieties over a field are "formally isomorphic" at a rational point (i.e., have isomorphic completed local rings) then they admit a common etale neighborhood. This is especially effective when trying to study algebraic singularities via their formal cousins (i.e., completions). So a framework for treating etale maps "as if" they are local isomorphisms (i.e., the etale topology!), even in the presence of arbtrary singularities, gives a way to pass back and forth between algebraic situations and formal situations.