Rheumatology is a specialty within the specialty of internal medicine. Their patients are treated mostly with medication and some physical therapy.
Orthopedics is a surgical specialty - and therefore their patients are all treated surgically.
There is some correlation between the two specialties. Rheumatologists will refer patients to orthopedic surgeons if the patient has, say, a knee all locked up with osteoarthritis and needs to have it replaced. Orthopedic surgeons who see patients with rheumatology-type symptoms will refer their patients to a rheumatologist.
Rheumatologists do not practice surgery unless they have been specifically trained to do so. I don't know any rheumatologists who have both specialties (and I know several of these specialists)
See the two citations below for more info.
No, they are quite different. An orthopedic surgeon diagnoses and repairs fractured bones in the human skeleton, and is a surgeon. A rheumatologist is most often a board certified internist (internal medicine physician) who is not at all a surgeon, but who diagnoses and treats chronic illnesses associated with the joints and the musculature, such a SLE (systemic lupus erythematosus, rheumatoid arthritis, ankylosing spondilitis, etc. Best wishes.