There's no definitive answer. It all depends on the airport and how their managment is structured.
In some cases you'll find that the ATSM is responsible for all of the air traffic services (controllers, engineers, briefing staff) with managers beneath him or her looking after each discipline. You might find the person looking after the controllers is called the SATCO. On the other hand, you could find that the person who looks after everything is called the SATCO. The only way you'll find out who does what is to see the management structure of the organisation.