A double gun has several advantages:
You have two chokes, so you can shoot two shots at a clay or live bird flying away or towards you, with the best chokes for the changing distance.
Cleaning is easier and not as critical.
Transport and storage are a pleasure with a takedown break-action gun.
A 30" double gun is still a good field length, whereas a 30" repeater borders on unwieldy; a 26" double gun is downright compact.
The warden factor: if you hunt, as I do, somewhere with "No Hunting" access trails, you can easily load and unload the chambers, and you never wonder if you forgot to put the magazine plug back in when you cleaned the gun.
The RSO, hiker and hunting buddy factor: it's easy for someone to see when a gun is open. It makes RSO's and hikers less nervous, and hunting with a buddy safer.
You can often get two different gauge barrel sets -- it's not cheap, but it's cheaper than buying another gun, and you only need to get one stock fitted.
You get two shots without having to pump.
You can easily check for barrel obstructions without a ramrod. Just break the gun open and look down the barrels.
Break-actions are much more forgiving of out-of-spec sized shells, and they don't scar up the brass. If you reload, you'll care.
But, a pump or semiauto has a list of advantages over a double, too:
Three (or more) shots.
Many inexpensive accessory barrels make it possible to have one gun for all imaginable purposes, including home and camp defense, big game hunting, small game hunting, waterfowling and upland hunting.
A gun can be tailored for many purposes with a barrel change, even if you're just fine-tuning for skeet vs. 5-stand, for example. You can make the same gun swing differently with a barrel change.
An 870 will do nearly anything, it's got good balance, and it costs a lot less money than a far less versatile double gun of equivalent quality and durability.
Gas-operated semiautos are "softer" feeling when they recoil.
Even the best double guns are a lot more delicate than a decent pump, or even a semiauto. If you know your gun will be heavily used and abused, get a pump gun. If you're going to lend your gun to a neophyte, make it a pump gun, unless you like the sight of your expensive break-action getting torqued.
Bottom line?
If you know exactly what you are going to do with the gun, and you're willing to buy an arsenal, a double might be your best choice. If you only want to or can afford to buy one gun, repeaters offer a lot of value. Even a lot of shotgun connoisseurs use a beat-up pump for ducks, because the guns are reliable and a lot less painful to drop in the mud.