能力 = potential energy; the energy to achieve your potential, i.e. the external force to achieve 本事 = your innate skill/talent/ability.

能力 is a specific skill of doing things or being excellent on all level. Eg. 1.动手能力，学习能力，赚钱能力，写作能力，工作能力... 2.他在这群人中是最有能力的。本事 1) implies how much money/power/resources/property we have. Eg. 她老公很有本事，在北京买了几套房子。2) rhetorical or usually used in a bicker or quarrel. Eg. 有本事了现在？(questioning the person who does something bad to you but not expect an answer) ; Eg. 你有本事你来（It is said when getting sick of someone being a backseat driver) 3) Stole Pen's example. 能做什么是谁的本事。

能力 is an ability that you can have it by learning. Like “英文能力” People who has ambitious for something then he/she makes effort on it. And ha/she achieves the goal call 本事 "他能當上CEO是他的本事"

本事：a little bit subject in Chinese, usually in disapproving or irony. Like, 你有什么本事这么说？ 能力: more objective. similar with CAPABILITY . when people "有能力做什么”，it means they are able to accomplish sth. usually approval.