In business, managing a team is usually a combination of leading the team and supervising the team.
The leader of a team is a member of the team and the first among equals; he or she works with the team to ensure that the team is collectively moving forward in the same direction.
A supervisor does not lead a team nor is part of it. With a business system in place, the supervisor works to ensure that everyone works according to the system and prevent any deviations from it.
A leader can motivate the team and has inside perspective. A supervisor can control the team and has outside perspective. Leaders unify; supervisors cause friction.
Managing a team is not an easy task. A good manager knows when to be a leader and when to be a supervisor.