I read Leaving Las Vegas before I saw the film and I was disappointed with the results. Nicholas Cage didn't encapsulate and inhabit the character. His portrayal of an alcoholic who has decided to drink himself to death was two-dimensional at best. We, as an audience, could have connected with the character even though he was not imminently likeable, but I found that I was waiting for him to kick the bucket at the end, instead of hoping that he may change his mind and live on.
Having said that, the script was not up to much and the direction was awful, especially the unbelievable plot device of having Sera's character spilling the beans to a "counsellor" in order to facilitate the narrative exposition.
Read the book, it's much better.