The differences are both very simple and rather complex. One of the things that adds to the confusion is that Martin isn't consistant about what they build - but lets try to summarize the differences as I understand them.
First, lets forget scale length for a while and start with gut strung guitars back in the 1800's and early 1900. They were classically (pun intended) 12 frets clear of the body and basically a size very close to what we would call a double ought (00). Here is a Hauser copy classical and my 000 - you can see the similar body shapes
Martin, Washburn and the others who were starting to build guitars "braced for steel strings" built similar bodies - 12 frets clear and 0, 00, and 000 sizes. When they built the first OM they wanted to give the players better access to the frets around 12 so the kind of pushed the upper bout down to the 14th fret (or in the case of Gibson, the 13th on a couple of models). They didn't move the neck, they simply shortened the body. You can see that when you compaer my 14 fret OM 12 string with the 12 fret 000
Notice how the upper bout on the OM has been flattened down to the 14th fret. The effects of that change on sound are probably pretty subtle - in theory you would have a slightly smaller air volume and a slightly reduced surface area of the upper bout (which probably has very little affect on the sound).
In theory also, you could build either body style with any scale length you want - start your layout at the 12th fret (whether it is at the body joint or not), make the neck the right length to put the nut at 1/2 the scale length, and locate the bridge where it needs to be. In practice, using a longer scale will tend to move the bridge more into the center of the lower bout (on a small body like an 0 it moves it past center). As JJ says, that has a big affect on sound - think of hitting a drum head in the center or closer to the rim. The other affect of scale length is string tension of course, shorter scale has less tension, is easier to fret and bend and some people feel has a bluesier sound. Longer is probably better for downtuning.
Going back to the OM 12 and 000 sixer - the 12 is short scale and the 000 is long - you can see the difference in the location of the bridge/saddle. To carry that to an extreme, here is the OM 12 once again along side my very long scale (26.2) 000 12 string. The body on that was built in the same mold as the 000 six, but the the scale length is almost an inch longer. That one was designed from the beginning for very phat strings and very low tuning - it is currently 4 half steps down from concert (C) and has a roar unlike anything else you'll ever play
You can really see how far the bridge has been moved down. Notice also that the necks are almost the same length but there are less frets and greater distance between them on the one on the right (some pretty long pinkie reaches on some chords).
Like I said, Martin confuses the heck out of things by combining different scale lengths, neck widths and body sizes - here is a summary of some of them
http://theunofficialmartinguitarforum.y ... topic/5309
And last, Martin, Gibson and others have built some great 12 fret dreads - they are fondly known as "slope shoulder" - they frequently have wider necks with slotted headstocks - the D28S, D12-20 and 35 and JJ's beautiful work are good examples.