Listen to the folks at your local brew shop.
I imagine re-using corks is possible, but I would never risk my wine to save on a 25 cent cork. Glass can be cleaned and sterilized easily, so reusing wine bottles is fine. Re-using corks is not. They are porous and have a million nooks and cranies that can harbor bacteria, so cleaning them is nearly impossible. Also, they've already been compressed for months, if not years, in the first bottle of wine. They may not fit as tightly in the second bottle.