I wasn't there for New Years or anything, but we arrived January 6th (2004). The marathon was that weekend (umm Jan 10 & 11). We really felt the parks were empty. And we didn't really see any more people in the parks during the marathon weekend. Our longest wait was for the House of Innoventions and that was 20 min (I don't know if we timed everything correctly or what). But we only used fast pass like 3 or 4 times. Most rides were under a half hour wait. And a lot we rode continuously (Splash Mountain, Space Mountain, Big Thunder Mountain Railroad, Primeval Whirl). It was a good time to go, except the weather. It was predicted to be in the 70s, when we got there it was in the 40s and 50s. So if you do go, pack hats and gloves (they are hard to find in Disney World and when you do find them, they cost an arm and a leg). My sister says she will never travel another time of the year because she liked the low crowd level. I agree (except we are going early December in hopes to get the Christmas stuff and warmer weather).
I don't know if the marathon is the same every year. But it seems like its the first weekend of January (except for when we went cuz New Years was the first weekend).