Well you guys have already covered the main points (the AI, atmosphere, etc.). I would also add a few more things, however--things that maybe aren't as quite as central to MTW, but still add to its appeal:
1.) Eras - I think most of us agree this is one of the game's most brilliant features, and a major reason why it's so replayable. The ability to play in the Early, High, and Late period allows people to try various strategies, tactics, and units. It allows each of us to play in a style that suits us.
2.) Titles - The ability to bestow titles on your generals (and removing them, as necessary) adds a nice bit of strategy, and can really affect the affairs of your empire.
3.) "Hero" generals & agents - We get real-life generals such as El Cid, William Wallace, Strongbow Fitzgilbert, & Joan of Arc; and also real-life agents such as Thomas Beckett (a Bishop) and Guy of Gisbourne (an assassin). All these characters definitely add flavor to the game, along with a nice touch of realism. (Yes, I know that Guy of Gisbourne was probably only fictonal. Silence!)
4.) Civil wars - How cool is it that you can actually assassinate/burn your own king, and then choose to back either the Loyalists or the Rebels? Or deliberately drive down the loyalty of a high-ranking general and get him to lead a rebellion against the crown (and with any luck, he'll sieze the throne for himself)?
5.) Units - In MTW, units are more intuitive and (at least partially) more realistic. Many troop types get a bonus from being trained in a particular province (Longbowmen in Wales, Vikings in Norway, etc.). Also, a number of units can only be trained in certain areas (desert archers only in North Africa & Middle East, Spanish Jinnettes only in the Iberian provinces, etc.). If you retrain a veteran unit to replenish its losses, the unit's average experience goes down, reflecting that it had to incorporate raw recruits into the regiment.
6.) Mods - I'm sure that in time, Medieval 2 will spawn a host of different mods for it. In the meantime, however, MTW already has a ton of them available for download. Whether you're talking about mods that expand on the medieval setting (BKB, MedMod, XL, etc.) or total conversion mods (Hellenic TW, Pike & Musket, Napoleonic, Samurai Wars, etc.), you're almost guaranteed to find a mod that enhances the original game and/or greatly extends its replayability.
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