Animal imagery is rampant in Brave New World. Just look at the first chapter. There's the repetition of "straight from the horse's mouth," Foster's implicit claim that "any cow" could merely hatch out embryos, the platitude that "Rams wrapped in theremogene beget no lambs." Later, when John goes to the hospital, he sees the Delta children staring at Linda with "the stupid curiosity of animals." The hordes of identical bokanovskified twins seem to him "maggots." It looks like Huxley's message is clear: the new world has so dehumanized its citizens that they now resemble little more than animals. The irony is that "civilization" should seek to elevate man, to make him less primitive, to put some distance between him and the other creatures of the world.
Animalistic traits really come into play when it comes to sex, probably because that's one of the basest, most universal instincts. John even quotes the "goats and monkeys" line from Othello, delivered when the hero imagines his wife copulating with another man the way that animals do. Also, Mustapha's response to John's comment – "Nice tame animals, anyhow," is brilliant (on the part of Huxley, not on the part of Mustapha). While John is disgusted by the bestial nature of the new world's promiscuity, he misses the purpose behind it: animals are tame. Animals can be controlled. In this way, the people of the World State are like pets – not like free people.
But it gets really interesting in Chapter Eighteen, when the crowds come swarming to see John standing around whipping himself for having dirty thoughts. The descending helicopters are described as "locusts" and then "grasshoppers" – fits with what we've seen so far. But it soon becomes clear that, while John (and, the tone seems to suggest, Huxley as well) condemns the civilized folk for being animals, they view him in much the same way. They throw food at John as though he's an animal in the zoo. (Huxley makes this explicit for us with the phrase "as to an ape.") This explains why they take pleasure in his suffering: because they can't see him as a person. To them, he's just animalistic entertainment.
This raises an interesting question for us: of the savages and citizens, who is more human, and who more animalistic? The notion of suffering seems to have a lot to do with this. John tries to prove his humanity by inflicting pain on himself. Clearly, no animal would revere the soul over the body enough to do so. It seems likely, then, that John's suicide is the only definitive way to establish his identity as a human being and not as a creature.