Javed: I cannot tell you anything about price. First of all, I would direct you to this short explanation, as it has some pictures and explains the difference between the two locks:
https://britishcountrysports.wordpress. ... explained/
The short explanation is this: Sidelock guns are essentially based on lock designs dating back three centuries or more. Think of the lock and exposed hammer of an old flintlock gun, like the British Brown Bess musket of the latter 1700s. Take a mirror image of the lock and place it on the other side of the gun, and add a second barrel.
This is what the double barrel shotguns of those old days looked like. When percussion ignition came along, the frizzen and flint-holding hammer were dispensed with and replaced with a percussion hammer and nipple for the cap.
When the era of breech loading was entered, the barrels were hinged to the action so that they could be loaded from the breech (Unless we're talking of the French Darne shotgun, of course!
)
The final evolution was to make the hammers smaller and place them inside of the action, in an area inletted into the wood stock. This is what sidehammer shotguns essentially are today. There have been refinements, but this is what you have.
Box lock shotguns, as the article explained, were developed in conjuction with breech loading. They are simpler and usually cheaper to make. The entire mechanism is contained in a more compact receiver.
As the article explains, the sidelock mechanisms are mostly located in the stock wrist area. This makes the wrist of the gun slender and sleek. The box lock, on the other hand, is "thicker" and more bulky, so the wrist of the shotgun isn't as trim and slim. Often for hunting quick upland birds, a light, trim sidelock is preferred, because it "swings" to the target more naturally that a box lock.
Sometimes, a box lock will have fake sideplates added to it, because of the sidelock's "snob value," or, there will be raised panels of wood behind the box lock that suggest a sidelock design.
Fine guns were made in both styles. The question would then be, what do you want to use the gun for? If you are a collector of fine guns, you would probably go with a sidelock, although there would obviously be exceptions for fine Greeners and such. If you wanted a light weight straight stock gun for upland birds, you might go with a sidelock, also.
If you are hunting waterfowl and quick swing isn't as important, or shooting trap, a box lock might work well for you.
Consider the article that I've linked and we can go on from there.