From Garrett Reisman:

'Mission Specialist' was a designation we used during the Space Shuttle Program to refer to astronauts who were not either the Commander or Pilot. So Mission Specialists did not directly operate the flight controls on the Shuttle, but we did serve as flight engineers, robotics operators, payload operators and spacewalkers.

So 'Mission Specialist' was a bit of a misnomer since we were really generalists - trained to do all those things. Only once you were assigned to a specific flight did you specialize and train intensively on the tasks you were to perform on that mission.

Today NASA considers all astronauts to be simply 'Astronauts' and there is no longer a distinction between 'Pilot Astronauts' and 'Mission Specialist Astronauts' as there was back in the Shuttle days.