The difference is everything, the difference is nothing.
It depends on whose perspective it's coming from. By definition, all plastic surgeons are trained and capable of performing "cosmetic surgery," but not all Cosmetic Surgeons are Plastic Surgeons. What does this mean? It means that the true education, training, and ability of a Plastic Surgeon encompasses ALL that is cosmetic surgery, but those who call themsevles "cosmetic surgeons or cosmetic doctors" are not all Plastic Surgeons. The reason is that there are many different groups of doctors and even nondoctors, who would like to have the mystique and therefore the "confusion" of cosmetic surgery or plastic surgery next to their names without the true training of a plastic surgeon. In other words, there is the "real deal" and then the "wannabes."
This distinction is extremely important if you consider the importance of education, training, and ability of the person who will be making permanent changes to your face, hands, breast, body, etc with such instruments as a scalpel, electrocautery, or a suction cannula. Much of what is done in surgery is irreversible and the best chance for success is always the first chance. Why would you risk your face to someone who might have taken a weekend course to call themsevles a "cosmetic doctor"?
You might ask, "So what is the big deal? So what if a family practice doctor or a chiropractor says s/he is a 'cosmetic doctor/surgeon'?" The problem is that those who don't know the difference will likely think that this "cosmetic doctor" is a plastic surgeon. Next thing you know, you've just had your face or body irreversibly botched up by this person. It's about safety and real credentials. If you had trained to be an elite in the special forces of the military, a Navy SEAL or an Army Green Beret, you would not take kindly to someone military or nonmilitary stating they were something "like a SEAL or Green Beret." You either are the real thing, or are not. You either took the difficult, real path to the destination, or you did not.
There is only one plastic surgery board that is sanctioned by the American Board of Medical Specialities (AMBS), the gold standard of medical specialties, and it the American Board of Plastic Surgery (ABPS). For those of you who are embarking on the path to become a plastic surgeon, you need to know this difference and understand what it takes to become a part of an elite specialty where only 200 true plastic surgeons are produced per year at major academic medical centers. You can choose to join other boards with "plastic or cosmetic" in their names, but they are not true speciality boards. Some of these boards do carry some weight, but there is still only one that is part of the ABMS.
So back to the question. It depends on who you are. If you are a real plastic surgeon, the difference is everything; if you are not, you'd like to say the difference is nothing.