I think a problem with this hard, and, I think, simplistic, definition (rape is power and control/sex is not), from the perspective of the person who is deciding whether they are raping or having sex,* is that if they ask themselves whether they are exerting power and control over someone or pursuing sexual pleasure, and if they conclude that they are simply pursuing pleasure, they may feel they can't therefore be raping someone, which is not true.
*Pro tip: if you're asking yourself this question, you need to stop whatever you're doing.