You could of course start from any topological group G acting continuously on a topological module M and define the cohomology H*(G, M) by means of continuous cochains. But what is sought actually is a "cohomological functor", i.e. a machinery which transforms a short exact sequence of modules into an infinite exact sequence of coh. groups. If your group G is profinite and your module M is discrete (this is equivalent to saying that M is the union of all the submodules fixed by the open subgroups of G), then the above construction works perfectly, what you obtain is "Galois cohomology", which is essential e.g. in infinite Galois theory. But in number theory, some deep problems require a profinite G (e.g. a Galois group) acting continuously on a compact M (e.g. the p-adic integers), and here you no longer have a coh. functor.