This is a question that I get a lot. Many people, maybe most, assume that these descriptions are all interchangeable and there is definitely a lot of overlap. Music therapists will tell you that in order for music therapy to be administered, there must be a trained music therapist present to help the patient select music they love and then to present that music for them and, if possible, process with them afterwards. With music medicine, the music itself is actually the entire intervention. This is very useful when a music therapist is either not on staff, or just not available because of patient load or other responsibilities.
When I have worked at hospitals in the past, I conducted numerous music groups each week, and depending on whether I was working with geropsych patients, chemical dependency patients, or new mothers of preemies, there was specially chosen music for the benefit of that population with specific goals and measurements. For me, music medicine is usually music with people who are ill in mind, body or spirit, as opposed to music healing or music wellness.
For the average person, music is enjoyed off and on throughout the day, just purely to lift the mood, calm the spirit, energize the body or just to entertain themselves. Music can be used by everyone to recall happy times, celebrate special occasions and speed the day along. Trained music therapists are important and can sometimes help people that no one else can reach or help, but everyone can benefits from the joys of music!