Crust is basically metal with drumming that sounds like Motorhead's. It's rarely technical.
Grindcore is a fusion of death metal and hardcore punk. I think it was first played by punks, but then you have Repulsion, so that's something that can be argued until the end of time. It's generally considered that Napalm Death and Fear of God started it. Napalm used to be a peace-punk band (check their earliest demo) and Fear of God came from a punk background as well. To my eternal shame, I have no idea about Repulsion or Carcass.
Powerviolence started out as a more extreme version of hardcore punk (No Comment being the best example of that) without any metal influence, but that changed pretty quickly. The easiest way to describe it would be to say that it's grindcore with stop-on-a-dime changes into slow, sludgy parts. Personally, I find that as long as it's not aping another band too much, it's more interesting than grindcore. And sometimes more violent.