There are three major differences between the novel and the film of Fahrenheit 451. First, the film is set in a contemporary near-future, without many of the dramatic science-fiction elements seen in the novel; the Mechanical Hound and the wall-sized television screens are absent, as are the nihilistic children and the jet-powered cars.
Second, the war, which is a major plot point in the novel, is barely mentioned; while in the book it was a source of concern and provides the climax, in the film, it only appears in a throwaway line in the background.
Finally, the character of Clarisse is changed dramatically. In the book she is a very young girl; her influence on Montag is through her ideas, and is killed halfway through.
"There was a girl next door," he said, slowly. "She's gone now, I think, dead. I can't even remember her face. But she was different. How? How did she happen?"
Beatty smiled. "Here or there, that's bound to occur. Clarisse McClellan? We've a record on her family.
You ask Why to a lot of things and you wind up very unhappy indeed, if you keep at it. The poor girl's better off dead."
(Bradbury, Fahrenheit 451, Google Books)
In the film, Clarisse is a teacher, trying to affect the minds of her students through unconventional teaching methods, and becomes a love interest for Montag. Instead of dying, she escapes the city with Montag and they start a new life together. This goes against the novel's themes of entropy and humanity striving against the constant grind of time, but Bradbury himself enjoyed the changed premise.