The American Paint Horse is a specific breed. Most registered Paints are also cross registered with the American Quarter Horse Association or AQHA. The breed itself is based on the Quarter Horse, although there is such a thing as a Thoroughbred Paint as well. (Paint TB's don't race, and aren't accepted as purebreds by the Jockey Club.)
Pinto, on the other hand, is a non-specific term that refers to a horse with a spotted coat. Horses with this designation can be of any breed. There are Warmbloods that are pinto horses, and there are also numerous other breeds that have pinto coloration.
The difference between the terms "Paint" and "Pinto" is comparable to the difference between the American Albino, which is a recognized color breed, and a true albino, which is caused by a genetic mutation and results in a horse's having pink skin, white hair, and clear or pale eyes. True albinos are relatively rare- I have known only one horse with this mutation in my entire life. That particular horse was a mare that I rode while in college- she was a school horse there. She turned out to be an albino Quarter Horse.
Perhaps this clears up your confusion.