Cognitive science is no more nor less than Behaviourism finally granted the permission to theorise about the internal causes of behaviour. Skinner is just conflating mentalistic explanation with ANY form of internalist explanation. That's brain science. But it can also run, jump, see, hear, categorise, manipulate, talk, etc. So it is behavioural scientists who must take on the task of explaining the brain's behavioural capacity.
Cognitive science includes behavioural and cognitive brain science.
What makes anything cognitive is that it explains in terms of internal structures and processes.