Although the engine was tweaked so that MTW looks a little bit better, both games are still essentially the same from a visual/graphics standpoint. The main difference between the two is that MTW features more options (especially on the campaign map) and greater complexity at the cost of less-than-optimized balance and AI. I would also say Medieval's atmosphere isn't quite as superb as Shogun's (although it's still very good IMO).
MTW has more playable factions, more political/diplomatic intrigue (trying generals for treason, dynastic marriages/alliances, civil wars, etc.), and a more sophisticated & involved religious model/system (Crusade, Jihads, Inquisitions, ex-communications, etc.). Your
dramatis personae -- kings, princes, generals/governors, etc. -- are more personalized now, as they have ratings for their Influence/Loyalty, Dread, Piety, Command, & Acumen, as well as possessing various Vices & Virtues which affect their abilities on & off the battlefield. Finally, Medieval includes a Glorious Achievements mode as an alternative to playing your standard Domination/conquest game, which -- while not perfect -- offers the player an alternative way to beat the game without having to conquer the whole bloody map.
All that said, there are a number of things in which MTW is noticably inferior to its predecessor. The AI (on both the campaign and battle level) isn't quite as good as in STW, probably due to the game's increased complexity. There are fewer "chokepoints" on Medieval's map -- which makes it harder for the AI to cope with the larger strategic situation -- plus the game has a lot more "hybrid" units (troops that are proficient in both ranged & melee combat), which the AI doesn't really know how to handle. In addition, Medieval's AI seems less able in its ability to decide when to go to war with other factions and/or backstab its allies; many of us MTW players talk about how our 20-province superpower was attacked by a 2-province kingdom.
Also, as
Wasp already pointed out, Shogun is much better balanced that Medieval. There's no "uber" unit in STW -- pretty much every troop type can be countered by another -- nor is any clan unwinnable (although the Hojo & Uesugi nearly always do well, of course). MTW has quite a few units which are either over- or under-powered, and the same can be said of some of the game's factions -- the French & Egyptians nearly always become superpowers, for example, while the Turks & Aragonese are often eliminated inside of the game's first 50 turns.
Finally, while Medieval has a great atsmosphere, Shogun's is still unquestionably superior. The artwork, the sounds, the voicework, and the music/soundtrack all combine together to really draw the player into feudal Japan and the world of the Samurai. When I play STW, I really *feel* like I'm there leading my clan to greatness (or ignominy, if things aren't going so well

). MTW does a pretty good job of sucking me in as well, but to a lesser extent.
So to sum up: Medieval has deeper, richer, more complex, and more dynamic gameplay than Shogun does. It also has greater replayability, since it has "more" of everything: units, factions, etc. In conrast, Shogun is a relatively simple game, but its elegant simplicity is part of its charm & appeal: It has fewer factions & units, and is much better balanced. In addition, the game's AI and overall atmosphere IMHO remain unrivaled by any of the other Total War titles, MTW included.
In the end, you can't really go wrong either way. Both are excellent games in their own right, and both have their strong points & weaknesses.
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