The most important similarity between these books is:
Their dystopian future setting: Both books are set in the future of the modern age. Technologies and sciences have progressed, while individual intelligence is kept at a specific, controllable level. The technologies of the future are used to control the citizenry, rather than to further human progress. Instead, the overpowering government uses its influence on culture to limit the amount of individual thought and reasoning ability in its citizens; one important shared theme is the banning of books. The citizens of Brave New World are knowingly conditioned to hate books, and view them besides as a waste of time:
"Back to culture. Yes, actually to culture. You can't consume much if you sit still and read books."
(Huxley, Brave New World, huxley.net)
More than just a waste of time, the possibility of outside influences could disrupt the carefully controlled society; for example, John's knowledge of Shakespeare is seen as an aberration and the citizens are encouraged to mock him.
The citizens of Fahrenheit 451 are similarly conditioned, but they do not have the advantage of understanding their conditioning. Instead, they think they are vastly superior than their forebearers:
"...I've always said, poetry and tears, poetry and suicide and crying and awful feelings, poetry and sickness; all that mush! Now I've had it proved to me. You're nasty, Mr. Montag, you're nasty!"
(Bradbury, Fahrenheit 451, Google Books)
There are other similarities as well: the protagonists are loners, isolated either immediately or over time from society; citizens are kept content by appealing to their baser desires and instincts; important or distressing issues (such as war) are kept from the public eye; the government uses spies and informants to weed out dissent; the higher-ups in the government allow themselves to break the laws, but prevent similar behavior in their citizenry.