Both disciplines have historically been at each other's throats, and Radical Behaviorists like B.F. Skinner often completely reject cognitive psychology at a philosophical level.
It seems that today Behaviorism has largely fallen by the wayside but certain principles like Operant Conditioning remain entirely valid.
Are these two philosophies/schools of thought incompatible or part of a greater whole? Is Behaviorism still relevant?
I think cognitive scientists would say that these views are compatible, insofar as cogsci admits results from behaviorism as valid results to be explained by understanding the cognitive constructs underlying them. Obviously (they would say) we have minds, our minds arise from physical processes in our brains, and as such have internal states that sometimes explicitly manifest in behavior, but other times do not manifest in any way that is perceivable. This is not a problem for non-behaviorist cognitive scientists.
Hardcore behaviorists, however, cannot be so accommodating. Internal and non-manifest mental states are not admissible in their theoretical universe, and so what most cognitive scientists would have no difficulty in describing as a state change in some internal mental edifice, behaviorists must reframe as some kind of 'mental behavior' in the same way they tried to do with 'verbal behavior.'
Of course, if one is very rigorous, and pushes very hard on this behaviorist view, it is either rendered nonsensical, or becomes an excessively ornate re-statement of normative cognitive science in behaviorist language. What this accomplishes I have no idea, although, to be honest, this seems to be nearly a straw man at this point. Hardcore behaviorists are increasingly hard to spot in the wild.
