To answer your questions about GPS heading:
1. you have to be moving for GPS to show any direction, if the car is parked there is no way for GPS to know where the car is facing.
2. There are actually 2 types of North: magnetic and true North. As the name would suggest Magnetic North would be North as shown on magnetic compass and point to somewhere in North Canada at the moment (it slowly changes) and it's different for different parts of US. The difference between true North (pointing straight to the North Pole) and magnetic North is called declination and for continental US can be between -20 to +20 degrees depending on location.
As you probably already guessed, Compass will always show magnetic North.
With GPS it is preferable to use true North, however magnetic North could be calculated using correction tables as long as you know your location. I would imagine your discrepancy is due to one showing magnetic North and the other showing true North unless your electronic compass is not calibrated and adjusted especially inside all metal car. I hope this helps.