They usually just call them roadcases. I dont think the differentiate names between padded and non padded ones. They are great because they hold everything you could possibly need including effects, so you have your whole rig there for you without setting it up all of the time. The con is that when you fill it up, it usually weighs a metric ton, and can be a bitch moving it to places where they dont have ramps.
My main setup:
Ibanez uv7bk green dot, Engl fireball 60, genz benz g flex 2x12.