Pistol grip stock vs standard stock for tactical shotgun:
Note: Pistol grip stock means a stock that comes with a pistol grip. I'm not talking about shotguns with a pistol grip only. Those are not recommended because they are generally designed to be shot from the hip, are hard to shoot accurately, and put a lot of recoil in your hand.
Pistol grip stock pros:
Many of them are collapsible M4 style stocks with adjustable length of pull, making them easier to store, and making them easily adjustable for different shooters.
Some, like the Knoxx Spec-Ops stock have recoil reduction buffers.
If you shoot AR-15 / M-16 rifles or other rifles with a pistol grip, the ergonomics are similar.
You can keep your elbows lower and tighter with a pistol grip, which is safer for going around corners without getting your elbows shot off. With a standard stock you have to have your firing arm's elbow cocked up like a chicken wing.
Standard stock pros:
With a Mossberg, it's a lot easier to manipulate the safety which is mounted at the top of the receiver. If you have a pistol grip Mossberg, you have to release your firing grip to use your thumb to manipulate the safety. I do NOT like the pistol grip on a Mossberg, it makes manipulating the safety very slow and uncomfortable.
Generally sturdier with fewer pieces to break.
You can carry the shotgun held muzzle up in your hands(with a pistol grip stock, it's very uncomfortable to carry muzzle up). Not sure how much of a pro this is since I think carrying muzzle up is a bad idea for safety reasons.
You can hit someone with the buttstock easily(not that I think this is a good idea). With a pistol grip stock it's pretty much impossible.
Overall I prefer the pistol grip stock, assuming a crossbolt safety(e.g. Remington 870). If the gun has a tang mounted safety(e.g. Mossberg) then a standard stock is much preferred.