The confusion for me comes from page 46 of Goldstein, where he says
"However, it has been proven that no such varied path can be constructed unless [the differential equations of constraint] are integrable, in which case the constraints are actually holonomic."
In the second edition, he gives a citation in a footnote. He continues:
"A variational principle leading to the correct equations of motion can nonetheless be obtained when the varied paths are constructed from the actual motion by virtual displacments."
I have consulted other numerous texts (Fetter and Walecka, Lanczos, Corben and Stehle, Marion and Thornton, Symon, etc., etc.) as well as lecture notes from the web in an effort to get this straight in my head. If anyone can help clarify this issue for me I would greatly appreciate it.