Buckley was not a Neo; but his writing, especially "Up From Liberalism," gave a new shift to the Moynahan, Lieberman types who fit the classic definition: liberals who moved to the center because of discomfort with what had happened to liberalism in the '60s, and then kept moving to the Right. The "Young Americans for Freedom" group was an example of kids who were involved in the migration and the anti-sds reaction. I was one in college. Paleoconservatives, call them classic conservatives, if you will, included the klan, American nazi party, etc. The Neos had too many who while moving to the center and then right weren't moving fast enough. Neos aren't racists, war mongers, Gay bashers, etc. But, they remained enamored of too much 'around' the Constitution and not enough--although claiming to be strict constructionists, 'in' it. Hence, to many like myself, we made the logical migration to embrace the basic principle of conservatism in America: Constitutional rights. Thus, 'Constitutionalist.' Still holding to the principles of what is right and decent, but becoming aggressive defenders of what 'are' our rights and which must be protected.
In short, Neo stopped being relevant to those of us who were enthused about it as youngsters as something new and different from the old timers' conservatism, and needed to be replaced with something more centered in our most important beliefs as we became educated and matured. There are still Neos, of course, but they stopped evolving long ago. Their issues have become Israel, Iraq, Iran, monetarism, etc. I think there are more important things to be concerned about. Basic things like the BoR. They don't.