my two cents:
medieval has a greater unit selection, more interesting abilities for special characters (spies, merchants, etc..), the jihad/crusade mechanism is nice, castles and citadels are awesome, multiple recruitment slots are useful and free upkeep slots help a bit too
on the other hand, battle maps feel pretty generic and uninteresting, hovering the mouse on the unit you need on each city to find out when the first one will become available sucks, AI cheating when a faction is down to a couple of territories gets boring and adds noting gameplay wise, battles are quite arcadey, AI features tied to game difficulty means you won't see interesting stuff happening unless you also give them game bonuses too. also, lot of feature badly developed or missing when compared to the first medieval (no oil gates, map forts broken, other stuff too)
rome as a nicer campaign map, with better thought starting points for factions that keep each faction interesting and fun to play and better control points for strategic placement of armies, you can recruit any unit as long as you can pay them, balancing taxes is harder and city can lose money, maintaining order on larger cities is harder and require to actually plan out build orders, you can't build everything so you need to make decision and take commitment on strategies, influencing the campaign on long term and actually adding to replay value, army upkeep and retraining costs actually have game consequence so you'll find yourself use more of militia as front lines to avoid losing high value units.
on the other hand, roman factions are quite overpowered, it may be quite unfun to fight them on very hard with lame factions; AI is quite crude when it comes to attacking, a lot of features are just hidden and no one tells you about them (blockades nets you money, I think. maybe. can't remember if it was a mod or something), economy is badly broken (two nations with two region each trading generates way more income than a four nation state not trading, meaning that when you're big enough and no one want to trade you your economy is screwed)
and lot of other things