What are some comparisons between Fahrenheit 451 and Brave New World?

When we compare two works, we show how they are alike. When we contrast them, we show how they are different. This answer will focus on likenesses, as the question asks us to compare.

A chief point of similarity is that both future societies actively and aggressively work to eradicate independent thinking. In Fahrenheit 451, this is done through banning books. The government is willing to pay teams of "firemen" to seek out and invade the homes of people suspected of having books, in order to destroy the books. In Brave New World, children are conditioned through means such as tapes played every night and electric shocks to think exactly the way the government wants them to think. 

Both dystopic societies reinforce this central concept of numbing people's minds. In Fahrenheit 451 the government aggressively encourages people to get hooked on idiotic televisions shows. In Brave New World, the government uses constant activity, consumerism, orgies, and a drug called soma to keep people away from independent thought.

Both works have a male hero who rebels against thought control. Montag does this through getting hold of books, killing his firefighter boss, and escaping to a marginalized alternative society. Brave New World's John the Savage rebels when he is transported from his "primitive" society to the brave new world by clinging to Shakespeare, religion, and the idea that suffering is redemptive. He eventually kills himself.

What quotes highlight similarities between "Fahrenheit 451" and "Brave New World"?

This is an interesting question to consider because I have always thought the dystopian world presented to us in Brave New World to be a lot softer and less harsh than the grim dystopian futures of novels such as Fahrenheit 451. After all, if you rebel, you only get exiled, rather than brutally tortured or killed. However, one of the similarities that we can draw between them is how the future societies these novels present us with clearly try to dismantle the bonds of love and affection in institutions such as marriage and family.

Note how Brave New World has managed to do this and why they feel that getting rid of parenting altogether is a good idea:

What with mothers an dlovers, what with the prohibitions they were not conditioned to obey, what with the temptations and the lonely remorses, what with all the diseases and the endless isolating pain, what with the uncertainties and the poverty--they were forced to feel strongly. And feeling strongly (and strongly, what was more, in solitude, in hopelessly individual iolation), how could they be stable?

In this novel then, institutions such as marriage and the family have been dismantled because they involve unpredictable and uncontrollable emotions that go against the stability that the regime is so desperate to achieve.

In Fahrenheit 451, we can see that what comes between Mildred and Montag's marriage is the "family" that is beamed onto the walls that Mildred spends so much time watching. We see that in this world, media has deliberately interfered with and become more important than the bonds of love and affection that are so intrinsic to human relations. Note what Montag says about his relationship with Mildred:

Well, wasn't there a wall between him and Mildred, when you came down to it? Literally not just one wall but, so far, three! And expensive, too! And the uncles, the aunts, the cousins, the nieces, the nephews, that lived in those walls, the gibbering pack of tree apes that said nothing, nothing, nothing and said it loud, loud, loud.

We see how family has been superseded by a shallow, ersatz imitation transmitted by media that changes human relations and attacks concepts such as love and loyalty. For Mildred, her "family" is clearly more important to her now than her actual family.

Both novels are thus similar in the way that the future world that they present us with attack or dismantle institutions such as marriage and family.