Applications
Subsequently there were further technical extensions (for example in Godement's book), and areas of application. For example, sheaves were applied to transformation groups; as an inspiration to homology theory in the form of Borel-Moore homology for locally compact spaces; to representation theory in the Borel-Bott-Weil theorem; as well as becoming standard in algebraic geometry and complex manifolds.
The particular needs of étale cohomology were more about reinterpreting sheaf in sheaf cohomology, than cohomology, given that the derived functor approach applied. Flat cohomology, crystalline cohomology and successors are also applications of the basic model.
Read more about this topic: Sheaf Cohomology