Both books express and interest in the relationship between language and thought, books and life. In Brave New World, Shakespeare provides John with a language that allows him to understand the world in a way that citizens of BNW cannot possibly imagine because we imagine in words. There are more books in Fahrenheit 451, but they serve the same function, preserving and presenting ideas that otherwise would be lost.
The major difference is Huxley's interest in and concern about science and the modern world. Coming from a family of scientists with a special interest in biology, he was afraid that a scientific bureaucracy would control the world in the future, using manipulation of the population to achieve "painlessly" what Orwell did through force in another modern dystopian novel, 1984. This is easy to see in the opening of the book. Reproduction, the most natural of phenomenon, has been replaced by scientific reproduction that allows the creation of individuals, designed by the state, to create a maximum of stability and a minimum of consciousness/creativity. The journey through the hatchery is reminiscent of the journey through the birth canal, although the end product is totally different.
I believe that this is the main difference between the two books should you wish to explore it. Whereas they both deal with the role of language/books in life/thought, BWN is much more concerned with the complicated role of science in our world and who controls it for what purpose.
I'll start with the style of the books. Brave New World is much more satirical. There are times when Huxley seems to be enjoying creating this society. Look at the exclamations his characters let out, and the phrases they use. By contrast, Bradbury's novel is much more imagistic and poetic. Turning to the content of the societies, Huxley's society is much more completely designed, and extends further into the lives of the characters. They are shaped from before birth towards specific ends. By contrast, in Bradbury's novel, the social structure seems imposed on an older society: ours. There are still remnants of it everywhere, in the books most especially.
In both books, the new society is something we've done to ourselves, through laziness and a desire to have easier lives, full of mindless, uncomplicated pleasure - by way of addictive drugs or videos - without deep comitments or passions. (vs. Orwell's 1984, where the government or outside ruling force which has imposed the society on us) The language seems dated and the authors' images of the future seem quaint, but the central ideas and warnings as just as relevant today as when the books were originally written.