The rotary tumblers are all quieter. It seems to me, having both, that the rotary takes twice as long as the vibratorys do. The case mouths also get banged around a little more. A plus is that most of the rotary tumblers are designed for rock polishing. That means they seal against liquids so you can do a very thorough liquid media cleaning job if you want to? I occasionally do that with a little vinegar in the mix to fizz the primer residue carbonates out. Of course, I've also got a 2.5 gallon ultra-sonic cleaner I've been known to do that in, too. None of it necessary. I just like to play.
To dampen the sound of a vibratory tumbler, as the others have suggested, you need enough media in it. You need a tightly clamped solid lid. My Lyman's does that just fine, so I'm not sure what the problem with LHB1's Lyman was?
Anything you do that quiets a vibratory tumbler usually will also slow its cleaning action because it is damping or absorbing vibratory energy to lessen the noise. An open lid (only do this outdoors in s breeze away from where kids play because of all the lead-containing primer dust that will be in the air and on the ground near the machine) is fastest and by far the loudest. The lid has mass, so that reduces the vibration level a little. The floor gets loud because the vibration transmits through the feet which try to keep the unit in place. If you take a carpet remnant with medium shag and turn it upside down and set the vibratory tumbler on it then very little will transmit through the pile to the floor. Instead the carpet will vibrate on top and the pile will act like little wobbly legs or as a sliding surface that let the carpet scrap move. Unfortunately, vibrating the carpet scrap consumes vibratory energy too, so it gets quieter but cleans more slowly.