No. Casual is not "purer" than Advanced. How does one even define purity when the act of role playing is literally defined in its own name? Role playing = Playing a role. It's akin to acting, which is in turn just professional level playing pretend in imagination-land. It's why Dungeons & Dragons was defined as role playing: A humongous portion of your entertainment value was literally playing your role, was getting into your character, and so on. However, role playing is not squared off to just Dungeons & Dragons acting skills--role playing is essentially just playing pretend. So if you're really into a video game, like Left 4 Dead, and start playing [i]as[/i] Bill--watching your team mate's backs, always using the M16, leading the way--you're role playing. It's just a different form of it, no more or less valid or pure than any other, and to say otherwise is pure elitism.

Now, yes, there is actually a definable skill to role playing. You can see the difference between a terrible actor and a professional actor, for example: One gets into the role, adjusts their appearance, the tone of their voice, the look in their eyes, and [i]really[/i] adopts what it is they're doing, while the terrible actor just imagines himself as person X, he doesn't actually really "become" person X, he's always consciously separate from it and never gets into the role. He fails to play his role.

The same applies to forums. You can see the difference between someone who gets into a character, builds a back story and tries to show multiple angles of that character, versus the person who's playing a walking, talking author insert, or a person who's field of comprehension of literature and role playing is merely tropes and [i]how[/i] they are used, not [i]why[/i].

Casual and Advanced, while different in attitude about the type of writing used, do not in any way differ about the format of role playing used--PbPRP, specifically. Neither one is more "pure" than the other, it's skin-level differences. What you're more likely seeing is the boisterous voice of advanced proclaiming advanced literary knowledge and not understanding that they probably [i]don't[/i] actually have that knowledge just because they can write ten paragraphs per post, and then confusing that with the skill of role playing.

Summary.
To put it metaphorically: Casual and Advanced are different in artistic style, [i]not[/i] in that one is art and the other is not. If Casual is to games what Advanced is to literature, they're both still art, and they're both still role playing, in equal parts. Advanced just tends to get the two confused more often because it tends to get filled with more wannabe professional writers than Casual does.