A physician who is a DO (Doctor of Osteopathy) is held to the same standards as an MD. They both attend four years of medical school and complete their training during the same residency programs. A surgeon who wants to be board certified in a specialty will take similar tests for certification, regardless of whether they are a DO or an MD. The differences between a DO and an MD stem from differences in teaching styles and focus during medical school. Osteopathic schools have long instructed medical students in looking at the whole patient, rather than the injured or diseased part.
While this was a major difference in the past, MD programs are also embracing the "whole person" approach, so this difference is not as notable as it once was. Both schools are now educating future doctors to be aware of the patient as a whole person, not a set of symptoms.
DO students are also educated in Osteopathic Medical Treatment (OMT), which is body manipulation similar to that which is done by chiropractors. Not all DO physicians use OMT in their practice.
For example, imagine a patient who goes to his family physician because of chronic headaches. A patient who has a headache who goes to an MD will likely to be examined and tested thoroughly for medical reasons for a headache, while a DO may include manipulation of the neck in his evaluation.
In the end, you should choose a surgeon based upon his proficiency with the procedure you need, his willingness to answer your questions thoroughly and the willingness to work toward your goals as a patient. The DO versus MD issue is really a non-issue when selecting a surgeon.