The squad characterisation and ambient dialogue are just some of the best ever written.
The cast are just genuinely funny, and whilst the campaign does get a bit 'hoorah' by the end... it never takes itself too seriously.
BC1 might have the funnier story (a bunch of screw ups decide to steal some gold (Kellys Heroes)) but BC2 is probably the better game.
If in doubt youtube some of the 'meet the squad' adverts EA put out for BC1&2, or watch some videos of the background dialogue between characters.
BC2 is still the best damn one, everyone has the same handicap anyway. Shame it took them until what, was it BF3 or 4 where they finally put out high tickrate servers?
Edit: To add to why it's so good: Building destruction! Still the best destruction of buildings in the entire series. Yes, the scripted pre-canned destruction in BC3 and 4 looks cool, but overall it's a step down from a last gen game and that's such a shame