With digital capture, color response is predictable, so you get accurate but boring images straight out of camera. Highly saturated colors, highlights and shadows can also clip because you don't have enough bits, so there's less room for error.
With film, color response varies by film, ISO rating, light source and exposure [1], so each one has a very particular character ("Kodachrome look", "Provia look", etc.). Also, analog media isn't prone to clipping, so you can get deep reds, deep blues, detailed highlights/shadows and still have the rest of the scene perfectly exposed without a lot of work.
It's hard to achieve the same characteristics with digital in post-processing because the limitation is at the capture step. It's like trying to fix a dull audio recording - no amount of work in post can bring up a frequency if the mic hasn't captured it.
[1] http://www.kodak.com/global/en/professional/support/techPubs...