Giles,
For a declared emergency we roll all the rescue equipment, provide priority (number one, for whatever you need) service, which means that I will move other aircraft out of your way, make you number one for the airport, and do whatever is needed to get you on the ground. Normally rescue equipment is staged around the runway that you intend to land on and they chase you down the runway after you land. Once an aircraft with an emergency lands the airport (or that runway) is closed until they check the runway for any debris, this can take a few minutes or longer depending whether the aircraft was able to clear the runway.
For a medical emergency, depending on the nature of the medical emergency, and your request, I will provide priority service that will enable you to land as soon as possible at an airport that is suitable equiped to handle your medical emergency. The airport rescue department will normally only roll their ambulance or dispatch medical personal to meet your aircraft at the ramp or gate, this is why it is important that we know the nature of the medical emergency.
The basic difference to me as a approach controller or tower controller is that the medical emergency doesn't close the runway or airport, and you normally taxi to a parking ramp or to the terminal. Both in my book are emergencies, one is a problem with the aircraft, the other a problem with a passenger.
Mike
NATCA FWA