Leading is the art of "moving" people - to action, toward a better future, in pursuit of a worthy goal. Managing is the act of moving "stuff" - inventory, products, services, fleets, etc.
Some leaders are also excellent managers, but know they can't possibly do all the managing, too. They learn to juggle managerial responsibilities, with the healthy cooperation of many others, while continuing to stay focused on the big picture tasks of leading.
Some leaders are not so good at managing, and realize it. Consequently, they figure out how to compensate for their managerial shortcomings by effectively enlisting the help of others in that work.
Some leaders are not so good at managing, but don't know it. The organizations they lead are often "on fire," lurching from one crisis to another. The whack-a-mole game we play at the carnival comes to mind with this type of leadership model, with tremendous effort being expended in reactive bursts, and more frustration than success to show for it.
Finally, some leaders are excellent managers, so they try to do ALL the work of leading and managing. They find themselves over scheduled, sleep deprived, family starved, and over worked. Despite their considerable gifts and talents, they often crash - physically, mentally, and/or emotionally.
Wise leaders constantly assess the state of their own wellness and that of the organization. Cooking either is a not a useful outcome.