Same relationship as "Coke" and "soft drink". You realize that, don't you? Bushmaster is the brand, and AR-15 is the type of rifle. .223 is the caliber it is chambered in.
I realize that the Bushmaster uses an AR action as do many things similar and otherwise dissimilar to an AR-15. The AR-15 is a specific model manufactured by Colt after the rights were sold to them by Stoner. The media has, however, taken to calling pretty much any semiauto rifle or occassionally even shotgun an AR-15 so as to link it to an M-16.