Some people can become confused with the available job titles and the type of tasks that they need to do depending on their position. When you say manager, this is someone who would need to focus on the external factor that may target the department.
A manager is expected to have a higher order of management skills because even the supervisor is under the manager. The supervisor will be in charge of handling the issues that the internal side of the department may have.
The supervisor is expected to have a great knowledge of the tasks that the employees would need to do every day.