What are the differences between a "trauma center" and an "emergency room"?

Ability to treat. An er is a location to triage and attampt stabilize a trauma patient before they are sent for definitive treatment. A trauma center is where the patients are actually treated and managed. There are several levels of trauma centers, up to and including the level one trauma center where there are surgeons and anesthesiologists present for 24 hours a day to take care of these patients.
Different. Er is just for emergencies and stabilize patients until they are routed to appropriate level of care. A trauma center is a hospital has a group of highly trained physicians and surgeons taking care of trauma patients. These include specialties like general surgery, orthopedics, neurosurgery, ENT etc.