NFQHA is a much newer registry (founded in 1995) to preserve "foundation" Quarter Horse lines.
AQHA was founded in 1940 to promote a specific type of horse, which included a body type (stock type) and coloring (no "excessive" white). The first horses registered in AQHA came from a combination of backgrounds, including Thoroughbred, Arabian, Morgan, and even grade horses. These horses were picked because they matched the desired phenotype. After the breed was established, AQHA closed the registry to outside breeds, except Thoroughbreds, who were allowed to register as "Appendix Quarter Horses." There are, as a result, some AQHA Quarter Horses that have a significant amount of TB in their bloodlines.
The NFQHA strives to have bloodlines including as little of that later TB influence as possible. To register a horse with NFQHA it has to have 80% or more foundation bloodlines. This is calculated by tracing back the horse's pedigree 10 generations and subtracting for each registered TB found. (More details