Having played both games extensively as well as all their official expansions, I believe I'll have to go for both
Seriously, here in Flanders you can get Rome + Barbarian Invasion for five euros, Medieval 2 without the expansion for ten. They're both absolute steals and can both keep you occupied for hundreds of hours.
However, I do believe you should play Rome first - simply for the fact that Medieval 2 takes the formula and tacks on a LOT. Better visuals, better sound, a bigger campaign map with more settlements, the papacy which has a huge effect on how you're going to wage war, more agents, and best of all, an expansion pack with FOUR different campaign maps each with their own variations, large and small, on the original campaign.
Countering all of that, Rome has but a few areas where it truly exceeds Medieval 2. First of all, the factions of Rome have more variety between them. In Medieval 2, for example, the tech trees of the Western European factions have a lot of similarities; Milan, France, the Holy Roman Empire all have Peasant Archers, Sergeant Spearmen, Dismounted Feudal Knights, etc. etc., whereas in Rome, Gaul has a radically different tech tree from Germania, which have a radically different tech tree from the Scythians, which have a radically different tech tree from the Parthians... You get the picture.
Secondly, Medieval 2 is a bit more buggy when it comes to unit controls. Pathfinding issues in sieges are bloody annoying, but most of all, cavalry are a pain to use - you order them to charge, they lower their lances and gallop, then a few metres before their target they slow down to a trot, lift their lances and walk into battle with their cavalry swords. Le sigh. Not even the patches solve this.
So I'd recommend you start with Rome.