The chapters are connected, yes, which each campaign following on from the last, but the game is no more story-driven than its predecessor was.
As before, there are little details in the maps like the scribbling on the saferoom walls and such to provide a bit of atmosphere and there is a bit of subtext hinting towards things, but there's nothing really concrete. What attempts that are made to hint towards a deeper story are all very unobtrustive and can easily be ignored if you are so inclined.
I speak as someone that loves a good story in a game, and actually thinks that a good story is one of the most important things for a game to have (besides gameplay, obviously), but the L4D series does not suffer in the slightest for not having a deep story. In fact, I'm pretty sure it would detract from the experience that Valve have aimed to provide with the games. And that is the pure and simple fun that comes with getting from A to B, cutting a swath through hordes of zombies as you go.