The 61 & 62 A body may have had a slightly better construction, but I suspect the difference was more likely NASCAR's safety requirements changed a lot between the late 50s and the early 60s, and has been growing more concerned about driver safety ever since.
Early auto racing was a deadly sport. It was because of all those early deaths the rules began to place more concern on the wrecks being more driver survivable. Better built cars, and better race track safety were required to keep racing, and new rules forced those changes.