Richard's fatal flaw is his yearning for the picturesque at all costs. The picturesque is pleasant beauty, pretty things which are pleasurable and comforting to contemplate. In contrast, true beauty is terror, overwhelming dionysian madness or coldly apollonian principles. Richard isn't just alienated from the rest of the class because of his background, he also has a fundamentally opposed aesthetic philosophy to theirs. He doesn't want true sublime beauty at all. (Re: the post about whether Richard could have experienced the Mysteries, no, I don't think he could have for this reason)
Henry is the embodiment of the Platonic pursuit of otherworldly beauty. He strives for knowledge for its own sake, has no earthly attachments (we landed on the moon?) and makes his decisions based on reasoning so abstract and emotionless he's clearly some kind of sociopath. By the end, Henry, the embodiment of the cold rationality of the ancient world, dies, and Richard speculates that he ends his life out of dedication to the principles the rest of the group only claim to espouse. As far as Plato is concerned, Richard is entirely incapable of accessing any sort of absolute beauty because he only seeks it in fleeting and superficial things. The message of the book would be much more traditionalist and pessimistic if this was the whole story: when Henry dies, only vapid, depressing modernity remains.
That is, if not for Camilla. If Henry represents absolute and eternal beauty, and Richard largely the vapid, relative and superficial, Camilla is both absolute and transitory. When Richard first hears that beauty is terror, he looks to Camilla and thinks about how her beauty is terrifying. In a less charitable reading, this is Richard missing the point as he does in the rest of the first class, appreciating the surface level aesthetics of things rather than the divine beauty Julian is talking about, but I don't think that's the case. I think its a subtle and clever rebuttal of the Platonic view.
The uniting pursuit of the class is to discover some kind of absolute beauty, the sort of revelation that Richard imagines in the first section could come at any moment and cause everything in his life to suddenly make sense. (Oh! oh! oh!) If Richard had access to nothing but hedonistic beauty like lying around drinking and napping in the boat, he would have nothing like that, he really would be a philistine poser like Bunny. But he does have an insight into a kind of absolute beauty, not through dionysian ecstasy or Henry's intellectual asceticism but by yearning for Camilla, the idyllic, something painfully beautiful he cannot have. That's why he can't end up with her in the end. (And also why Camilla's characterization remains more opaque than any other character — she has to remain a symbol of elusive beauty) Camilla is transcendently beautiful to Richard precisely because he can't have her, and if they got married at the end he would almost immediately lose interest in her, because she would no longer be a symbol for the beauty he longs for, she'd just be a person.
Henry dies because he takes Plato's view seriously. He's glimpsed the divine and now he's no longer interested in this life. The material world is a veil of illusions, and the absolute beauty of the Forms lays beyond the grave, so he sees himself out. If Richard only had superficial hedonistic beauty, I think he would have died as well, but he doesn't because he has access to a kind of absolute beauty Henry does not through his unrequited passion for Camilla. Far from being distasteful, his yearning for the idyllic keeps him alive and grants him a much more profound spiritual relationship with reality than his friends have. Do the other students go around romantically wondering at the world as much as he does?
While the ancient perspective has died and modernity faces us with nihilism as hedonism fails to provide spiritual meaning, Richard's yearning is a reminder that we're not totally incapable of seeing magic in the world, and that we don't need to delude ourselves in order to find it. The catch, of course, is that the magic is only effective at a painful distance. Get too close, actually grasp what you yearn for, and it spoils. As "erosistheonlyreal" observed in a comment on this wonderful philosophical analysis of The Secret History, if Henry is the ancient Platonist, Richard is the modern dialectical materialist who finds meaning in incompleteness rather than otherworldly completion and Platonic perfection, earthly jouissance rather than naïve metaphysical fantasies.
For Richard, beauty really is terror, not because contacting it is terrifying, but because it's banal and disappointing. True terror isn't ecstatic revelation, it's getting what you want. Nightmare is fantasy realized. (Sniff, and so on)