What subdivision algorithm advances have occurred since Catmull-Clark? In 1978 Edwin Catmull and Jim Clark defined the recursive subdivision process that bears their names, and although those principles are applicable still today, what advances have occurred as far as optimization and accuracy?
More of an extended comment than an answer:

What do you mean by "optimization and accuracy"? Do you mean computational efficiency for some particular application, like ray tracing, physical simulation, CAD modeling, ....?

Note that the idea of "accuracy" for a subdivision scheme is not well-posed. Different schemes will have different limit surfaces, but there is no canonical way of declaring one limit surface to be "more accurate" than any another. One can pose some constraints on the type of limit surface one desires, but these constraints are again very application-dependent: one person might ask for $G^n$ everywhere, the next will complain because this precludes preserving sharp creases.

Catmull-Clark (and Loop, for triangle meshes) remains popular because of its simplicity, which in many cases outweigh its weaknesses (no handling of sharp features; loss of regularity at extraordinary vertices). Countless alternative schemes (which may or may not be improvements over Catmull-Clark, depending on the specific application) have been proposed -- if you have a specific application in mind, with specific requirements, we may be better able to help you navigate your options.