Well, you CAN do both residencies if you really wanted to, but I doubt someone would want to do all that. It is true that you can work in most Emergency Departments after surgical training, since most EM residency programs are relativiely new - so many ED's in this country are staffed by docs from all specialties, not just board certified EM docs. EM programs would like to eventually change this, but right now there is quite a shortage of residency trained EM doctors.
To answer your orignal question, an EM doctor does a lot more medicine, not just surgical work. A surgeon can also handle medical problems, but usually they only work with patients who might need surgery or already had surgery. So in the ER, a EM doc will evauate patients with pneumonia, MI's, strokes, as well as coughs, colds, headaches.
In a trauma situation, both EM and surgery usually work together, but the surgeons usually have primary responsibility for evaluating whether the patient need surgery, putting in any invasive central lines, chest tubes, etc. EM physicians more often control the airway (anesthesia does this in some hospitals too). ER docs 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.
I had the big dillemma between EM and surgery. I chose surgery, mostly b/c to me it seemed the surgeons get to do all the cool stuff in trauma. Ok, there's more to it than that. Another HUGE difference I see between the two is that in the ER there is not much long-term follow up. ER docs 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 (especially in an oncology case) and developing that long-term relationship. 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.
Another big difference: surgery is a longer residency with a lot more on-call time and much more demanding hours in general.