The rivalry first occured simply because critics couldn't stop comparing Bad Company 2 to Modern Warfare 2. This was true of everyone. Critics, fans, and even video game awards, it was always a case of BC2 vs MW2. And MW2 would always win, with critic scores, video game awards, and direct comparisons on places like IGN.
Despite that, because the internet had a mayor hard on for hating on COD, and BC2 was fresh and new, everyone started a war. Had you asked back then, most would've replied that BC2 was objectively the better game.
Not to mention BC2 itself took some jabs at MW2 during the campaign, though they were tongue in cheek.
After the war was over, MW2 ended up winning basically in terms of sales, and everyone went on their way. There was a problem however, when marketing Battlefield 3, EA was EXTREMELY agressive towards Call of duty. And not tongue in cheek like in BC2, but full on, 'our game is better' kind of agressive.
The marketing phrase was 'above and beyond the call', ffs.
They even compared the situation to Rock Band and Guitar Hero, saying something along the lines of "Nowadays, you don't see Guitar Hero on stores selling as much as Rock Band"
He was literally saying "Our game is better, and it's going to sell more"
This caused a massive war between fanboys of each game. Some claiming that BF3 was going to be the second coming of Jesus FPS Chrsit, and some claiming that MW3 is simply more fun and that BF3 was all graphics and not much else. It was a shit flinging contest which I took part of on the side of Battlefield. Man, I regret every second of it.
It ended up backfiring though, as some poor decisions such as Battlelog (don't get me wrong I love battlelog but back then people were FURIOUS about that shit) and being Origin exclusive (No steam! Back then that was fucking ridiculous, specially since Origins was not a tenth of what it is now) lost them the battle.
MW3 ended up selling 6 million first day, while BF3 was around 5 million the first week. Obviously still great, but it definitly didn't beat COD like they claimed they would.