In reply to:
I second the question…what exactly do they do? What are the basic advantages over simply programming a flight plan into GPS?
Sorry. I was going to answer the second part of your question but got distracted and forgot. The only thing a flight-director has to do with GPS is that it “follows” whatever navigational signal you are tracking (satellite based or not). It also gives attitude cues. Basically, the flight-director is a visual cue of where the autopilot would be placing the aircraft to perform a given manuver. During a required change in course, for example, the flight-director bar on the attitude indicator will turn. If the autopilot is on, it will follow the bar. If the autopilot is off, than you need to follow the bar in order to stay on course.
An FMS or Flight Management System is a completely different thing and usually has GPS built in and an output to “drive” the autopilot through complex maneuvers (along with other features like NAV/COMM tuners and performance databases).
Here’s a link to answer your question about flight-directors.
Here’s a link to an example of an FMS.