What is the difference between a council-manager system of administration and a mayor-council system?

The fundamental difference is in who has the legal authority to hire and fire city employees. In a council-manager system, the manager has that power. (There are limited exceptions. Under G.S. 160A-173, city attorneys are appointed by city councils; and in some council-manager cities, the city charter provides that the council appoints and removes the city clerk and, very occasionally, the chief of police.) In a mayor-council system, the council collectively has the power.