Regarding the Sheitlach mentioned by 613. She is most likely quoting from a Chabad posek. In Chabad they are makpid to go with a wig and not with a tichel. They hold a wig is more efficient (and so did Rav Moshe) then covering with a tichel.
Chabad is the only place I know that has a halachic preference for wigs. An interesting question is what to do if a Chabad woman goes to a place where they pasken that wigs are halachicly prohibited (and no woman there wears only a wig in public.)