It depends somewhat on the region you're talking about. Different hospitals allow different things.
An EMT (as I think you're using it) is an advancement over the basic life saving class taught by the Red Cross. Initial training runs about 150 - 200 hours. Basic goal of the class is to take the patient from the scene to the hospital and provide some care during the process. A lot of the care involves getting vitals as in emergency situations the changes in vitals can be indicative of the problem.
Care can include administering oxygen, bandaging, splinting for transport, CPR, insertion of some airways, and assessment of the situation.
I'm assuming that you're referring to a Paramedic as a medic. It is used as a term, although it more likely to be used in the military. A paramedic (sometimes called a paragod) is also an EMT, but with a much more advanced level of training. Paramedics can insert an IT tube, administer some drugs, adjust de-fibs and shock in abnormal situations, insert IVs, and maybe perform some minor surgical procedures in an emergency.
The goal of a paramedic is the same of any EMT -- getting the patient to the hospital as fast as is safe in as good condition as possible.
It depends on the area how emergency medicine is set up. It can be under the jurisdiction of the fire department, the police, or an independent agency. Most frequent model is to pair an EMT with a Paramedic. (By the way, an EMT is technically an EMT-Basic. There are actually a couple of other levels.) Other places have Paramedics respond only in certain situations. Most medical calls only require basic skills.
Best of luck,
Jim Clark-Dawe