A multifamily community is a multifaceted entity and leasing (although very important) is just a portion of the daily activities.
Leasing is the act of acquiring new residents. A leasing manager would be in charge of all aspects of leasing and the leasing team. This would include prospect management, application approval, marketing efforts, etc.
A property manager is the CEO of the property responsible for the overall performance of the community which includes financial management, asset management, team management, etc. They do everything from paying bills and writing a budget to managing resident retention and most of all leading their team.