Cognitivism over Behaviorism
let's have a look at the social cognitive theory by Bandura:
It says, that behavior can be acquired by simply watching other people act on something. Maybe you have already heard of the famous bobo-doll experiment, if not here is a link:
https://www.youtube.com/watch?v=128Ts5r9NRE
Anyway: behaviorism doesn't care about anything going on inside our mind, like reasoning for example. It simply breaks behavior down to stimulus-response-mechanisms or behavior-response-interactions, that is why behaviorists call the mind a "black box". They simply think in a way of Input-Output mechanisms, the thing in between these two gets neglected.
So, going back to the social cognitive theory: Behaviorists can't explain behavior, that was acquired through observing other people. The person, who is the observer, has never experienced something like a stimulus-response trial, or acted on the behavior on his own. However, we know that it is possible and people need something like a memory, perception, reasoning, to learn in such a way. So, we need cognitive abilities to explain this kind of behavior (we have to look inside the black box), which is incompatible with behaviorism. 

In other words: every behavior, that needs or was being activated by something, that is a part of our cognitive abilities, is a behavior, that can't be explained by behaviorism.

Another example is problem solving: To solve a problem, we think about the situation, reassemble the elements and then act. Behaviorism can't account for such a behavior.