It depends on what kind of "carrier" you are talking about.
There is the breed English Carrier, which was one of the core breeds in developing the modern homing/racing pigeon. It's no longer used for homing, but rather a show breed. They are easy to tell apart from homers/racers, because they are tall and slender with very large nasal and eye ceres.
Then there is the "carrier pigeon" that was called that because they carried messages in WWI and II (and prior wars/events, of course). It gets on my nerves when people call homing pigeons "carrier pigeons" because I know that Carrier Pigeons are their own breed. The birds who served in the war were homing pigeons and racing pigeons.
The only thing that separates a homing pigeon from a racing pigeon is whether or not it has been bred for racing. They are the same bird, just one is bred based on race results, and the other is just bred if it comes home from where the person takes it. I guess it just comes down to what you intend to do with them. Think of it like horses. Say you have a couple horses as pets. Someone else may have the same breed and they use them for racing, breeding the better racers together in hopes of creating more winners. They are the same horse, they've just been raised and used in different ways, and the racing horse has a pedigree of ancestors bred for better results. Therefore, that horse may have better genetics for the competition. But it is still the same kind of horse.