Many, many things. How many bands have such raw talent and energy live? I've never seen a band get a crowd going like Paramore.
I also love, like a few others have said, their integrity. I think one of the reasons they sound so true live is that their records aren't messed about with in a studio.
Another thing? Respect and love for their fans. I still find it hard to believe that for a band as big as they are now, they're still giving fans the chance to meet them, and they still communicate with them. Bands like 30 Seconds To Mars? There's such an air of arrogance around them, whereas Paramore really aren't like that. It just astounds me how much respect they have for us guys, and in return, they get our respect.