Philosopher’s Stone doesn’t work very well with large handsizes, which is unfortunate because it’s otherwise always a good thing to improve your average handsize. Enter Vault: by allowing you to discard Victory cards and Coppers for $1 each, you can essentially make those cards count twice (once for Vault, and once more for Philosopher’s Stone). This allows you to continue using Tactician, Caravan, and Laboratory without worrying about any adverse effects on your cashflow.
If you don’t have a large hand, or if your cards are too valuable to be discarded to the Vault, then this trick isn’t worth very much. However, if you keep careful track of how many cards are in your discard and draw, you can selectively use Vault’s ability to squeeze out an extra $1 from the Philosopher’s Stone.
An extra benefit is that Vault plays well with Potions, because it can always convert them into $1 each if you have no need for an Alchemy card.