The radical thing that Skinner did was to admit mental states into behavioristic analyses, but in a subtle way. Those mental states were not called mental states; rather, they were included as a sub-category of the environment, and, further, limited to only one's personal environment. (Note that many radical behaviorists or behavior analysts would not agree to my categorizing the internal environment as a mental state.) So, in Skinner's view, each of us is affected by the external environment and by our own internal environment. But, the rules of engagement, as it were, are exactly the same in both, meaning that reinforcement, stimulus control, the three-term contingency, in short, all of the constructs of behavior analysis, apply equally well to both environments. Take the following example, given by Jay Moore (personal communication). You are playing tennis, and as you hit the ball, you say, "Keep your eyes on the ball." Then, you notice that the ball lands in the court more often when you do. So, you keep saying that phrase to yourself as you hit the ball, or, the effect, keeping the ball in the court, is reinforcing your private verbal behavior. But, consider what would happen if you said the phrase but the ball was just as likely to hit the net, the back fence, or the court. Then, that private verbal behavior would not be reinforced and, eventually, you would not say that to yourself anymore. Notice how it is the contingency, the fact that the phrase and the behavior affect each other, that determines the likelihood that the phrase will be repeated. Also, notice that the phrase itself has no power over the behavior, nor does it have separate status (ie. it is not a "mental event").
Behavior analysts are fun to talk to because they seem to live in two separate worlds. Like the rest of us, they have desires or wishes, but, deep inside, they know better. They know that those desires and wishes do not have power over them; they are just private environmental events to be reinforced or extinguished. In research, they are constantly on the lookout for contingencies of reinforcement. In some fields, they have been wildly successful. Look, for instance, at behavioral therapy and behavior modification.
Cognitivists argue with behavior analysts most when it comes to explaining complex behaviors. The issue is not whether a cognitivist or a behavioral analyst can provide an explanation; both can. Rather, the issue boils down to whether unobservables will be admitted as explanatory aids. Behavior analysts do not admit them while cognitivists do. Parsimony may ultimately decide between the two types of explanation, and some compromise may evolve. For relatively simple issues, behavior-analytic explanations may be more parsimonious. However, for complex issues, where behavior-analytic environmental constructions become laborious and unstable, then cognitive constructs may be more parsimonious. Time may tell.