From a guy who has played (beaten) and loved both games, lemme break it down for ya.
BF3 Pros: Much more 'authentic' feel to it. Laser sights and flashlights blind you, the sun is a real ***** sometimes, and unless you got a lot of rockets, some mines, or a jet/helicopter backing you up, that tank is gonna ruin your day. Getting into a dogfight online is some serious fun when you also take into context that while your flying up there, there's the ground war with other peeps down there. There is a ton of variety in the BF3 multi-player as well. Infantry fights? Check. Usable vehicles? Check. Playing as a medic or spotter? Check. Aerial dogfights and chopper-gunner overwatch? Check. Team-play is extremely rewarding (when it actually happens). The campaign was very good, though again, with a more 'authentic' feel to it, and while the story wasn't much to write home about, it was still good (Preferred MoH or CoD over it in story (subjective)).
BF3 Cons: When the teamwork aspect falls through, your game comes crashing down like a sack of bricks. If you want a simple frag-fest for some K/D brawling, it's likely to let you down. Since snipers are more viable (subjective) in BF3 due to the longer distance and overwatch positions, you can get ripped to shreds before you ever get to the fight. The campaign doesn't teach you crap for the equipment and vehicles you'll use in multiplayer, so expect a lot of trial and error when hopping online (Still can't fly a helicopter for shit).
MW3 Pros: It's Michael Bay's love-child. Explosions and epic stage pieces freakin' everywhere in the story. If you've even touched the campaigns for MW1 & 2, it's a very solid and satisfying conclusion. Sure it's surreal, but I admit that I had a big smile on my face through every level of the campaign. Spec-Ops is back with Survival Mode, which I unfortunately haven't touched yet, but intend to rectify that. Multiplayer is relatively the same as the previous MW games, though some nice changes have happened. If you're a sniper, quick-scoping's back. Weapons now have their own levels with their own perks. The level design seems pretty solid so far (Only done two matches, please don't kill me). Online is great for some simple frag-fest and getting your shooter on.
MW3 Cons: Let's face it, this is NOT how anything even close to real military actions take place, so please don't think it has an 'authentic' feel. Suspension of disbelief is mandatory for the campaign. If you intend to do the campaign, hope you've played MW1 & 2, where-as BF3 is standalone. The graphics engine is the same, though tweaked decently. Multiplayer is extremely similar to the previous few CoD games, with only smaller changes/improvements. Let's also face it, CoD has some of the most annoying tweens you'll ever play alongside/against (Thank god for Mute).
Misc Points: Both campaigns are relatively short, about 6-10 hours each. Both have Co-Op modes (BF3: Co-Op, MW3: Spec-Ops/Survival) and Multi-player. Each one's feel is unique to itself: BF3 has more authenticity, while MW3 heralds Michael Bay and offers you more boom. Both look very good, each with their own art style (BF3: Crisper graphics and longer view-distance, MW3: More color and a smoother frame-rate (And no blinding f***ing flashlights)).
TL;DR: Want team-work and authentic? Get BF3. Want hollywood boom-boom with a frag-fest and survival? Get MW3. Both are very good games, and both have their own unique ID.