Where are you located?

west tn

A slab is not usually a foundation. There needs to be a footing to support the exterior wall unless you are building on stable rock or the slab is 10 inches thick. The perimeter footing might be poured integral with the slab but typically it is the footing that is the foundation.

Would a "conventional" foundation be a perimeter wall with a framed floor over a crawl space or a framed floor over full basement?

Having lived on a slab for the last 30 years, I hope I never have another one. Despite being in FL, they are cold in the winter. They are also hard on you feet, legs, and back, since there is no give at all. Professional dancers won't dance on concrete. The other problem is servicing plumbing, and anything else under the slab. If a pipe fails, or you want to move things around, the only way to fix it is cut up the floor. With a crawl space or basement, you just work on it from underneath. They are cheaper- the only advantage I can think of offhand.

Macv, when someone refers to a slab foundation, they mean a monolithic combined footer/slab.
It's the preferred foundation throughout the mid-south region and into Tex. Wherever there is only a few inches of frost, and the terrain is table-flat, or can easily be made so. It's advantage is speed, minimal excavation, and economy of material. Cons are: second chances to fix or modify plumbing; house sits very low, less than a foot above grade; Only is aesthetically suited to ramblers/ranches. Two-story houses have a very squat appearance (not that they aren't built in spite of this).
Casey

I know how to pour an integral slab and perimeter footing but I don't know what the OP is comparing it to.

macv-i was questioning whether there was any advantages (besides cost) of building on a at grade monolithic combined footer/slab as opposed to having a crawl space.

thanks for everyone's input. we wanted a crawl space, but i was wondering if i was missing something wonderful about an at grade slab.

We built a crawlspace with a slab. We are in the country and have been plagued with the invasion of critters in the past. The crawlspace is dry, clean, and keeps the destructive rodents out. Decent storage space. Our furnace is there and DH has ease of access to run whatever lines, cables or pipes.