Strictly speaking, I think GPS is the satellite system and technology, and all it gives you is your position coordinates. Nav or Satnav is the device with the screen and the map data, that _uses_ GPS to show display a map and show you your position on it. But I personally call the gadget with the screen in my car "the GPS," the owner's manual calls it "the NAV system," and the last time we rented a car, they called it "Satnav."

No difference. I think people may use satnav more in the UK, but it would still be understandable in the US.

Same thing... Different Dialects.