What we understand as vampires and werewolves are revised versions of several separate folk tales. So the stories vary, depending on what werewolves and vampires can actually do, and what their weaknesses are. Good to remember. ;)
My guess is that silver is a valuable metal, and its purity works against the "corrupt" nature of vampires and werewolves.
I had a look, apparently the silver bullet story might come from the beasts of Gévaudan: these were apparently wolf-like creatures which terrorised France in the 18th century. The second one was shot and killed with silver bullets.