The first answer you've received covers much of what I would say; however, in the movie, Kate is the victim of a bet--pure and simple. She is "shrewish" because she's been burned before and when she finds out about the bet, she comes unhinged (understandably so). Bianca is the one everyone wants in the play, and while there is no "bet" placed in the actual play, Kate stands in the way of all of Bianca's suitors. The eldest daughter must marry before the youngest or face a life of spinsterhood. Therefore, Bianca's suitors attempt to arrange for Kate's marriage as a means to get to Bianca.

The movie is actually fairly closely tied to the play. Most of the differences are simple modernizations in order to make the movie "work" for the younger generation. Probably the most obvious of these differences is that the story is set around getting a date for the prom instead of getting married. Another difference is in the characterization of Kate. In the movie, Kate paints a picture of a rough relationship in the past that has lead to her recent disillusionment with boys and dating. In the play, we just see Kate as a naturally mean spirited shrew when it comes to dealing with people. One other main difference is in the relationship between Kate and her final mate. In the movie, Kate has definately fallen for the boy and we're lead to believe there will be a happily ever after. In the play, we don't know for sure if Kate is really in love (or even really likes) Petruchio or if she is simply obeying his will in order to avoid further taming.