The differences are thin and vary from situation to situation. There might be some situations where they are the same while in some others, they might be different.
Let's say a professor must have the capacity but he may not have the capability to undertake the assigned job if he has certain aspect of disability (such as too weak or too much energy required to undertake the job). In most of the scenarios.
In the relam of mental thinking,
Mental Capacity : The ability of a person to remember a lot of things simultaneously. Just like memory. ( His mental capacity is tremendous, so much so, that he can remember the entire word list only after one read)
Mental Ability / capability : the ability of a person to perform a specific mental task. ( Does he or does he not have the mental capability to do this task)
Mental Power : Shows the degree of difficult mental problems a person can solve ( He has immense mental power and can solve the most difficult of mathematical sums)
the words are usually interchanged and there does not exist a significant difference between them to warrant knowing the difference.
P.S. In my opinion :) I stand ready for correction !