While one of the products you mentioned may have been hydrated and the other
anhydrous, "glycerol" and "glycerine" are synonymous. "Glycerine" is the
correct name, since the -ol suffix should rightly be reserved for alcohols
(ethanol, methanol, isopropanol, etc).  The hydration state of the product
should not make a difference when treating periostracum, though it could
make a difference in other applications (like making nitroglycerine) :-)

Paul M.