Authority is the right to do something. Power is the ability to do something. And control is the actual doing of it.
For example, owning a car, and having the experience and skill to drive it have no impact on your life if you are sitting in a cafe. They are irrelevant to your experience of sitting in the cafe, drinking coffee, and maybe nibbling on a croissant. Owning the car gives you authority to drive it. Having had the lessons and the practice gives you the skill and ability (i.e., the power) to drive it. But without a car, or without actually being in the driver's seat, you have no control and you cannot experience it.
In BDSM it is the actual use of control---how it is used and how it is experienced---which causes the positive and satisfying reactions of many of its practitioners. Merely having the authority, or the skill and power, is not actually satisfying until control is taken and asserted.
In fact, while it's easy to focus on individual activities in BDSM---such as flogging, bondage, pain play, etc.---for many people who are in dominant/submissive (D&s) or master/slave (M/s)relationships, it is the exercise of power and control which provide penetration and engagement and, hence, the satisfaction and pleasure. For many D&s and M/s folk it is often the case that the actual activities they explore together are only important insofar as they create opportunities to use power and control. If one activity is unavailable, perhaps due to health problems or lack of equipment, another activity, which might seem entirely different, can be easily substituted as long as it provides an opportunity to use power and control.
For example, a flogging scene might create the opportunity for the dominant to physically manhandle or order around his submissive as he positions her and repositions her during the scene. This manhandling and ordering around are expressions of power and control and they can be just as intense as any pain or thudding coming from the flogger. This same sort of manhandling and ordering can equally be part of wax play where the dominant positions and moves around his submissive as he changes the locations where he drips the candle wax. Both of these very different activities can be just as intense and satisfying for both the dominant and the submissive due to the opportunities for physical or rough handling, and for command and obedience, even though flogging and wax play are very different beasts on the face of it.