Air mattresses are great and you could totally go camping with one, but there are a few drawbacks.
They're waaaay too bulky for backpacking.
They require an electric pump and hook-ups, unless you have an alternate power source.
A regular air mattress won't hold up as well over time if you're an avid camper.
Camping mattresses aren't just for travel, but are designed with that extra layer of convenience and durability for the outdoors, specifically.