By definition, the Public Service comprises the departments listed on the 1st Schedule of the State Sector Act 1988. Irrespective of being called a department, ministry or some other title, they are all Public Service departments.
By definition, the State Services comprises the agencies that operate as instruments of the Crown in respect of the Government of New Zealand (i.e. the Executive Branch of Government). This includes the Public Service, most Crown entities, the Reserve Bank, a range of agencies listed on the 4th Schedule of the Public Finance Act 1989, companies listed on Schedule 4A of the Public Finance Act, and a small number of departments that are not part of the Public Service.
By convention, the State sector comprises the agencies whose financial situation and performance is included in the Crown accounts as part of the Government reporting entity under the Public Finance Act 1989. This includes the State Services, tertiary education institutions, State-owned Enterprises and Mixed Ownership Model companies, as well as a small number of agencies that operate as instruments of the Legislative Branch of Government.