Well, the "caulk" type, as you call it, is mostly for use as a draft stop in tight spots around doors/windows or wall penitrations like plumbing or electric. Though I have seen it also used to fix the foam padding in a car seat. It's not the best for use as weather (water) seal, or pest seal since it isn't solid.
The other type I think you are describing is kinda like spray on rubberized undercoating for cars. It is used on thin metal appliances and such by manufacturers to keep them from sounding tinny and cheap, one common use is on stainless kitchen sinks. Never used the stuff, but if it IS like the undercoating spray, it'll just make a mess and stick to everything.