The profinite group cohomology of discrete modules can be defined by right derived functors. Its application includes Galois cohomology, Brauer groups etc. These facts demonstrate that defining profinite group cohomology different from the usual group cohomology is indeed meaningful. However, I don't understand why we only look at discrete modules in this case. Similarly, when considering profinite group homology, why do we only look at profinite modules?
I understand that, since profinite groups have a topology, it makes sense to look at topological modules that the groups act continuously. But why discrete or profinite?