I own both BF3 and MW3 on the PC, along with a number of the older titles from both series.
They're essentially two completely different games. The only relation is the fact both have guns.
I admire the look and sound of Battlefield. Tanks and Aircraft tend to be popular. I don't particularly like any of the maps that include aircraft however. The ones with vehicles are generally acceptable though. I like Conquest mode and generally only play it. I appreciate the squad element (even if it is annoying to be limited to just 4 players when 6 or 8 would be more useful for myself and mates). I like joining solid dedicated servers and playing the 500+ ticket type ones that allow long battles. I can appreciate the slower, somewhat tactical elements.
I admire the smoothness and superior movement in Call of Duty. It's just more refined and fluid. Simple stuff like climbing over objects is done better in COD. I like the variety of game modes. As mentioned I only really like Conquest in BF but in COD I can play about 5 or 6 fairly regularly. I like the fast paced games. I enjoy the choice of kill/point streaks.
I dislike the number of crashes in Battlefield. It's definitely improving as each patch comes along but it certainly suffers from more that it's fair share of cryptic "You've been disconnected" type messages. I had a complete crash earlier where the game just closed. No error message or anything. The game just disappeared. Battlelog is a bit of a faff. It could easily been implemented into the game rather than as a standalone in the browser.
I dislike the increasingly dated graphics in COD. I dislike the tendency of the developers to pander to consoles. I generally view the playing of first person shooters on consoles to be the height of idiocy so stupid pollution stuff from consoles like matchmaking and locked field of view on the PC is very irritating.
As you can see I like both and can see advantages and disadvantages in each game.
Which do I play more? Probably 60% COD, 40% Battlefield.