Screw drive is a lot quieter. If your garage is below a room, that might be a consideration. Screw drive also has a bit more upkeep - you have to lube the screw drive auger periodically, the chain drive ones you can pretty much ignore for almost ever. Also, with a screw drive you aren't as reliant on the springs. Screw drive can handle a lot more weight/pressure than a chain drive, so you can back off your spring, thus making the spring virtually fail proof.
I have a 35 year old chain drive that is starting to die, and I'm going to replace it with a screw drive very soon. My bedroom is over the garage, and that thing makes the bed shake like I can't do myself.
I never thought I'd make a post that included the word screw so often, and not have it be dirty.