There are a great many differences between the two games.
For example, Battlefield:3 is much more open. It is considered more realistic because it has a whole new level of game play. Most of your game-play environment can "blow up". There are vehicles and planes. The game has bigger maps and different game modes. Call of Duty on the other hand traditionally has smaller maps with some planes/helicopters but they are awarded through kill streaks.
In both games you can have customized classes however in battlefield you have 4 main classes that you can customize weapons geared towards that specific 'type'. Such as Engineer class will have sub machine guns mainly with repair tools and rocket launchers. While the sniper class would have sniper rifles and detection gear. Both games have weapon customization and each specific gun can be leveled up to unlock things like new sights for the weapons.
So to sum up:
Battlefield: 3 -Bigger maps, More 'realism', no kill streaks, tons of vehicle combat, and speicalization benifits. Typically slower paced gameplay.
Call of Duty: Modern Warefare: 3- Smaller maps, Kill streaks with each planes, etc. Less vehicles. Typically faster paced gameplay.
Planes in battlefield:
Killstreaks in Modern Warefare