If anything, Niles is the one who envies his brother, however mildly; despite his jokes about Frasier's "fast-food" psychiatry, Niles sometimes admits to envying how his brother is well-known, and that he helps many people over the air while Niles's own work is anonymous to the public.

I don't know where this idea that Frasier is a snob comes from. A "snob", to me, is someone who looks down upon those he views as his lessers; views them as fundamentally different and forever unable to achieve his own far greater status. Frasier never once acts as a snob to his uneducated father. He never shows anything but respect for Daphne despite being the product of a family from the bottom of Manchester, and never says or implies that she would in any way be unfit for Niles (Niles's own marital status conflicting with his lust is a different story).

Frasier sometimes gets irritated by Martin's habits, or failure to appreciate the same things as his sons. He sometimes gets irritated (but more often amused/bemused) by a rambling or bizarre story from Daphne. But if such is the definition of snobbery then we are all snobs. To put another way, we are all snobbish about some things (and the Crane brothers have more things on their own lists than most people), but only snobs believe that such things make them inherently better than others.