Pre-war Martins were built using Brazilian Rosewood for their back and sides, which is highly prized both for its look and its sound.
But because it's so beautiful, it was also heavily used in the furniture industry as well (I've seen some enormous conference-room tables made of Brazilian Rosewood, and they knocked my socks off), and as a result it was over-farmed.
So since WWII, the supply has been both rare and tightly controlled; as a result it's extremely expensive.
In the late 1960's, Martin switched to using Indian Rosewood instead, which is very nice, but not as nice as the Brazilian variety.
You can still get guitars made with Brazilian Rosewood, usually from small, independent luthiers, but they're very expensive.
Often times a luthier will offer Brazilian Rosewood as an upgrade option on a standard model, and I've seen the upgrade price as high as $1000.
Update : After having done some more research on Wikipedia , here's what I've been able to find out.
Martin developed the dreadnought-style body shape around 1916, but didn't alter it to accommodate a 14-fret neck until 1931.
Between 1931 and the war, apparently the craftsmanship was at its peak, especially when it came to carving the interior bracing.
During the war, materials and skilled labor were both understandably in short supply, so the quality suffered a little bit.
After the war
(I found this bit interesting), the guitars slowly started to suffer intonation problems, apparently because due to higher production, the jigs used to position the bridge gradually eroded and no one noticed, until they did and fixed it.
So there are some other non-Brazilian-Rosewood-related reasons why pre-war Martins are preferred over post-war, pre-1969 Martins.
But honestly, after having looked at a bunch of different sources, it seems as though the answer to the question "Why is a 1936 Martin preferred over a 1956 Martin, since they both used the same woods and design (and assuming no intonation issues)?
" is basically "Because people liked the way they sounded and were willing to pay a premium for them."
Which isn't exactly specific, and doesn't help you if you're wanting to re-create that magic with an independent luthier.
music.stackexchange.com/users/36/Alex Basson
Even more important to the tone of pre-war Martins than Brazilian rosewood is the wood species used for the guitar tops: Adirondack spruce.
Adirondack spruce comes from New England and southeastern Canada.
In the present day and for the past several decades, most guitar tops, and almost all Martins, are made of Sitka spruce, which comes from the Pacific Northwest of the USA, and western Canada.
There is a discernable tonal difference between Adirondack spruce and Sitka spruce.
They switched from Adirondack to Sitka spruce because Adirondack spruce became over-harvested around the time of World War II (it was used in building aircraft and aircraft parts, notably propellors) and billets of suitable size for carving guitar tops became unobtainable.
Adirondack spruce guitar tops are only now becoming available on a few new guitars, because apparently the Adirondack spruce trees are making a comeback.
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The dreads were used for bluegrass, and bluegrass guitar needed volume to compete with fiddle and banjo, and without amps, the only way to up the volume is to but heavy strings on it and play it like hell, so they started beefing up the construction so they wouldn't have so much warranty work.
For example, there's less shaving in the bracing and even a "popsicle brace" on more modern Martin dreadnoughts.
These are reasonable design tradeoffs, akin to the difference between the "stock car" in a NASCAR race and the stock car of the same type used on the road, and just as there are mechanics who will make a car ready for racing, there are luthiers who make their bones by taking off-the-rack Martins and making them ready for high-performance flatpicking, like the pre-war Martins.
I don't want to endorse any one luthier, but Dan Lashbrook is such a pioneer in the field that the process is called "Lashbrooking".
Of course, there's also wood drying out and all that, plus "people like pre-war Martins because people like pre-war Martins", but here are some documented and repeated differences.
music.stackexchange.com/users/195/Dave Jacoby
The wood is obviously important but the major differences are bar fret and no truss rod!!
It allows the wood vibrations to happen freely and generates the real Martin sound.
Tj Thomson fixes pre war MARTINS and also makes new ones with the technic.
The feel due to Bar frets is really harsh and not everyone can appreciate playing the axe built that way.
Lawrence Berndt, Wood god for most guitar builder in the US has come up with mushroom head bar frets and a T bar carbon fiber to replace the truss rod is coming out with a line of instruments for next Namm show 2014.
music.stackexchange.com/users/6711/Thianar Gomis
I believe that the pre-war Martin design positioned the top X-bracing forward about an inch as compared to the later versions, which moved them back, for a reduction in stressed leverage on the top, and corresponding reduction in warranty and repair issues.
The pre-war mahogany backed model(s) are as revered as the rosewood backed model(s), so the wood type is not a distinguishing factor in pre- vs. post-war Martins, IMHO.
music.stackexchange.com/users/24563/peter
Pre-war Martin guitars are revered because of their sound.  What differences are there between the construction of pre-war Martin guitars, and Martin guitars made now.  And are there any companies that are making guitars with any of these "old" features.  Edit:Looking for an answer that is more specific and comprehensive to the Pre-war martins.  Brazilian was found on guitars regularly until the late 60's, and still is found on some guitars, so that can't be the only difference. (or can it?)
How do Pre War Martin Guitars Differ From Modern Ones?
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S0: Pre-war Martins were built using Brazilian Rosewood for their back and sides, which is highly prized both for its look and its sound.   In the late 1960's, Martin switched to using Indian Rosewood instead, which is very nice, but not as nice as the Brazilian variety.  Even more important to the tone of pre-war Martins than Brazilian rosewood is the wood species used for the guitar tops: Adirondack spruce.  In the present day and for the past several decades, most guitar tops, and almost all Martins, are made of Sitka spruce, which comes from the Pacific Northwest of the USA, and western Canada.  There is a discernable tonal difference between Adirondack spruce and Sitka spruce.  The pre-war Martin design positioned the top X-bracing forward about an inch as compared to the later versions, which moved them back, for a reduction in stressed leverage on the top, and corresponding reduction in warranty and repair issues.   Many people believe that the pre-war use of Brazilian rosewood is a reason for the desirability of pre-war Martins. More modern production used Indian rosewood. Others believe that the change from Adirondack spruce to Sitka spruce for the guitar tops was more of an issue. However, given the reverence for all pre-war martin guitars, even mahogany backed models, the wood type if not considered to be a distinguishing factor by some people. In order to stand up to the heavier strings of the bluegrass players, later Martins were built more solidly, with stronger bracing. Their x-bracing was moved forward and there was less shaving of the bracing on more modern guitars. There are some companies and luthiers who specialise in taking modern martin guitars and converting them to the coveted older specifications.
Many people believe that the pre-war use of Brazilian rosewood is a reason for the desirability of pre-war Martins. More modern production used Indian rosewood. Others believe that the change from Adirondack spruce to Sitka spruce for the guitar tops was more of an issue. However, given the reverence for all pre-war martin guitars, even mahogany backed models, the wood type if not considered to be a distinguishing factor by some people. In order to stand up to the heavier strings of the bluegrass players, later Martins were built more solidly, with stronger bracing. Their x-bracing was moved forward and there was less shaving of the bracing on more modern guitars. There are some companies and luthiers who specialise in taking modern martin guitars and converting them to the coveted older specifications.