LongRifle's comments on pillar bedding are pretty good. I'd like to add a few of my own including what spawned the birth of pillar bedding.
Benchresters have weight limits for their rifles. And they love the heaviest, shortest barrels they can shoot 'cause they're very stiff. To add more weight to the barrel meant something else had to loose weight. Lee Six, a benchrest shooter of high quality decided a fiberglass stock would do as it would be lighter than a stock made from a tree. So he made some.
They did well except the cores in them was a bit too soft to hold conventional epoxy bedding. Someone figured out that metal tubes glued into the fiberglass core would withstand the mild recoil the small cartridges they used. And they wouldn't compress enough to worry about compared to the synthetic stock's softer-than-wood cores. The barreled action would stay in place for the life of the barrel. This worked well enough indeed. Match results often stated the winner used a "pillar bedded fiberglass stock" and the rage was on. Everybody had to have pillar bedded receivers.
Meanwhile, folks using wood stocks in the high power match rifle discipline had conventionally bedded them in epoxy. .308 Win. cartridge accuracy at 300 and 600 yards with this system from good barrels and ammo equalled or exceeded what benchresters got from their pillar bedded 22 and 24 caliber cartridges. As pillar bedding was contagious, some high power competitors thought it was the best thing to do. Many tried it, darned few equalled what conventional epoxy bedding produced.
Done right, conventional bedding works fine. Just torque the stock screws to what gives best accuracy (Win. 70, for example, 60 inch-pounds front and back screws, middle one to about 40 inch-pounds). Torquing pillar bedded receivers still compresses the pillars, but only microscopic; steel and aluminum do compress.
That pad under the barrel is often debated as to its worth. Few, if any, benchrest or high power match rifles win matches with an inch or two of bedding under the barrel's chamber. All that does is transfer any fore end bending (from position, resting it on a bag atop a bench or something else) to the barrel and bends it. The more the epoxy's out from the receiver, the more the barrel will bend from upward/sidways fore end bending. When fiberglass stocks first appeared on service rifles, folks leaving their rifles in the sun learned the heat warped 'em and their zero's changed. Bolt action rifles with completely free floating barrels didn't have a problem; only the stock around the receiver warped from heat ant not enoug to be noticed.
Wood moves with temperature, but not as much as those early synthetics. They're probably better now, though. If under-chamber epoxy-padding ever does help accuracy, it's my opinion that it's helping something else that's not right with the barreled action's bedding. I tried it three times years ago; rifles shot more accurate the more of it I removed so I quit doing it.