For me, it was all the great dedicated Rush maps, the accuracy of the guns. I think in the Bad Company games, you didn't have to lead your bullets like in the newer BF games, but it made the guns really fun to use. Some of the guns had the best iron sights I've seen in a game, while others looked really bad. Sniping looked and felt better. The scope would recoil back instead of up.
Things I didn't like the was how the guns would bounce around when you walked. I like the other newer BF controls more, but they're fine for a Bad Company game.
It's not better than BF3 or BF4, it just has a unique feel to the gameplay and guns that people like and miss. I want another Bad Company game with great graphics, Rush and Conquest dedicated maps.