What is the difference between immunology vs rheumatology?

Little! You have essentially two specialities that cover the fields. I trained in a division of clinical immunology and rheuamtology. Most of the non-structural diseases i treat involve the immune system. Allergist/immunologists will overlap, but the major focus is allergy and asthma than rheumatic issues.
Immune system. Immunology is a specialty that mostly focuses on deficiencies of the immune system, or diseases that make people susceptible to infections. Rheumatology, on the other hand, generally focuses on problems of an overactive immune system, whereby the immune system seems to attack itself (autoimmunity), in diseases such as lupus or rheumatoid arthritis.