Good Question because there is overlap between these 2 careers.
Both trauma surgeon and Emergency physician are in Trauma team and they both work in emergency department.
usually if there are trauma surgeon, orthopedic surgeon and anesthesiologist in the team, the position of ED physician is optional.
Emergency physicians are capable of many things like: surgical procedures, trauma resuscitation, advanced cardiac life support and advanced airway management.
Emergency physicians ideally have the skills of many specialists - the ability to manage a difficult airway (anesthesia), suture a complex laceration (plastic surgery), reduce (set) a fractured bone or dislocated joint (orthopedic surgery), treat a heart attack (internist), work-up a pregnant patient with vaginal bleeding (Obstetrics and Gynecology), and stop a bad nosebleed (ENT).
While trauma surgeons handles most injuries to the neck, chest, abdomen, and extremities (other than fractures which treated by orthopedic surgeons). Injuries to the central nervous system are generally treated by neurosurgeons. Musculoskeletal injuries are treated by orthopaedic surgeons.
So ER physicians get to do some procedures - lines, LP's, minor suturing, sometimes depending on the hospital, but they never get to do acutal surgery in the OR like surgeons do.
Another difference between the two is that in the ER there is not much long-term follow up. ER physicians work shifts, so they never have there own patients they follow up on beyone the shift. Surgeons, on the other hand, take a patient through from the diagnosis to treatment and to follow up care - often seeing the same patient for years.
Trauma surgeons even have this relationship - if they fix up a gunshot wound, they see the guy in clinic weeks and months later to make sure they are healing well.
By the way, here is Sudan they don't have emergency physician in EDs.