The group here has done a fairly good job of describing boxlocks and sidelocks. Some additional elaboration:
1.) The box lock was made by Westley Richards staff, Mr. Anson & Mr. Deeley. This was around circa 1880. The box lock specifically has only 3-4 parts per lock which makes for simplicity.
2.) The sidelock has enumerable patents upon it and many, many alterations. The most famous and currently duplicated rendition is usually the Holland & Holland 7-pin sidelock that has been used since off-patent by almost everybody.
To jump slightly off topic here, there are also numerous other components that have been proven the most reliable that are now used in almost all guns. The Anson Plunger and Deeley Latch are the two most ubiquitous methods for attaching a forend to a double gun (thanks again to Westley Richards & Co.). About 99% of double guns in the past 80 years feature one of those two options. The southgate ejector system has also been proven to be the most reliable and is used quite regularly in most fine guns.
I'd be remiss if I didn't point out that Boxlock and Sidelock are like Pepsi & Coke. Naturally, there are also other beverages out there and other lock systems also. Arguably the most famous "collector" lock technologies are the Dickson Round Action and the Mortimer Skeleton Action. Add to that the very famous Westley Richards "Drop Lock" which is not really a box lock at all but a gun that looks like a box lock with the ability to slide a button on the lock and watch the guts fall out the bottom in two separate assemblies. The novelty of the drop lock is easy internal cleaning and the option of having a spare set made in case you're in the bush and you have a gun failure occur.
As to value: The most valuable guns in history are sidelocks made in the UK between WWI and WWII by the "big 5" makers: Dickson, Holland, Woodward, Boss and Purdey. The guns were all off-patent by that point and the bits and pieces were the best amalgamation of parts that could be had with the most mature, expert workforce. (rapid downsizing from war and changing tastes left a very small gun trade with the cream of the crop of remaining gun workers)
As to owning something amazing: I think the most beautiful guns in the world are what I call "Patent Guns". From 1880-1910 every fine gunmaker was rushing to the patent office to express their ideas as to what is the best design for a gun. Some survived, some died, but in that 1880-1910 (the Silver Era as it's known) I believe much more interesting, personalized and beautiful guns were made and the latter "Golden Era" certainly had refinement but it rings as soul-less duplications of one another. Arguing the superiority of a Golden Era gun from Lang versus Grant versus Boss versus Churchill versus Woodward grows tiresome as they at that point were all so good and all so similar that arguments are purely collector arguments and aesthetics of a specific specimen. On the other hand, if you look at a best 1890s Westley Richards and a best Charles Lancaster and a best Charles Hellis and a best Stephen Grant, they all are stunning, uniquely made and wonderful different from one another. Oh, and the silver era guns might fetch 1/3rd of what a golden era gun of same make might fetch.
From an aesthetic standpoint, the sidelock gave the maker more room to engrave the gun which made them more often best guns. Boxlocks are not nearly as refined but there have been best box lock makers out there like Charles Hellis and Harkom to name two. (Westley Richards did not often make best boxlocks even though he owned the patent) . Then there was the attempts by the continentals and particularly the Belgians and Prussians that decided to use box lock actions and side plates so you'd gain the metal engraving real estate to beautify the gun but internally the guns were still boxlocks. (a modern example is the Beretta 687EELL for example..not sure why anyone would buy a modern one but that's my opinion)
As to durability of the wood that is up to debate depending on the wood how it was inlet and what the guts of the gun contain. A single trigger gun often had more wood removed from the head so that made them more fragile. (but that was regardless of lock type) Perkes ejectors required more wood removal on both stock and forend than a southgate system so that could be a point of weakness but again is regardless as to whether box or side lock.
Here's the real reason that gun snobs that know their stuff reject modern boxlocks over modern sidelocks. (emphasis being modern) Most modern boxlocks are affixed with a through-bolt holding the stock onto the locks. That is a great trick for quick installation and removal of a stock but it serves many problems:
A.) You can't easily cast and bend a gun that has a through-bolt (perazzi, krieghoff, berettas, all brownings, merkels).
B.) They are always chubby and inelegant because you have a large void in the stock to accommodate the bolt
C.) They are overbuilt
D.) Because they are bolting them together, they are reckless and imprudent as to which wood they select. The "best gun" wood that is used today on these through bolted guns is not of proper grain contour. (they should be angled straight grain in the wrist and burled or figured in the body of the comb)
E.) Throughbolted and bespoke do not often go hand in hand. At best, the stocks are slightly carved for gun fitting but not bent to fit the custom needs of a shooter.
Sidelock guns are never through bolted and that alone means they are of a more svelte, better built construction than most boxlocks. There will always be exceptions to the rules (and exceptions are often bargains).
Addendum: Wow did I ramble. My apologies!
, just the boxlock type will be cheaper. the westley droplock is another design where the mechanism can be removed from beneath the action
thats gotta 
