The primary purpose of snubbers on ground tackle is to absorb snatch loads. You use elastic rope
-- almost always nylon -- and this absorbs energy when the boat
moves back and forth with the sea and starts to snatch up. Think of it as a shock absorber.
Chain catenary may or may not perform this function. Mostly depends on how much weight of chain you have deployed and what the sea state is -- so the size of the chain is important. The problem is that when the chain starts to get pulled tight, the elasticity disappears. So if you "use up" the energy absorbing potential of the chain catenary, then you can have damage from snatch loads.
On my previous boat with 8mm chain, the snubber was absolutely essential -- the chain was too light to absorb enough energy to prevent snatching up in even in fairly moderate conditions.
On my present boat, with 12mm chain, I almost never use a snubber unless conditions are really rough and/or the water
is quite shallow.
Please do NOT use a snubber for Purpose 1 in Steve77's post above. A snubber is not designed for or suitable for taking the whole load of the ground tackle -- it's just a shock absorber. The chain should be belayed by a different, strong means -- preferably as strong as the chain (otherwise it forms a weak link). A strong strop or a strong chain lock is best. The snubber is no good for this purpose because, to be elastic enough to fulfill its primary purpose, it cannot be as strong as the chain. Also nylon, when working to absorb energy, is less strong in practice than its rated strength, due to internal heating
. On top of that, nylon loses strength when wet, and if that's not enough already, nylon is extremely vulnerable to chafe compared to other kinds of cordage. Your windlass won't hold your boat if a storm blows up -- don't leave your boat depending on an elastic snubber to stay attached to the anchor. Belay that chain!