When I think of power I think of someone or some system that directly influences the behavior of others. Some of the time power is used to influence someone in the wrong way due to arrogance. A lot of the time they see themselves as superior, above everyone. The people that are influenced by this power see them as self-conceited or egotistic. Power also seems to be a self-obtained attribute, it is a personal acquired trait. Kind of like seniority, you automatically assume that you have power because you have more experience or are older in age. Power resides with a person. Power has nothing to do with a flow of hierarchy, whereas authority does. Power is not authorized but, authority is vested by position. 
When I think of authority I think of power but in a formal sense. The individual with the author will use the power when necessary, for example: to give orders, make formal decisions, and enforce obedience. Authority can be seen as a formal right, a way to exercise power in a formal fashion. Given to those in high official positions, it follows a hierarchy. Those who have earned respect. The ones who have earned a high  official position. It is a way to exercise power in a formal fashion. Authority resides with a designation. Authority is given to a position whereas power is seen as inner strength. Authority can be seen as a delegated power but, power is taken by someone holding a position.