First, The Wolfman is a remake of another film by the same name made in 1941. Now, some of the Cullens, and possibly even Charlie, Billie and other characters from the Twilight series may have been around then, but author Stephanie Meyer was not. She was born in 1973. In addition, several werewolf movies were made before New Moon, including The Howling I-VII (The Howling VII, btw, is subtitled New Moon Rising and was made in 1994), An American Werewolf in London (and later, Paris), Wolfen and of course the grand-daddy of all werewolf films The Werewolf of London from 1935.
Second, and more importantly, the wolfman or werewolf as a preternatural creature has existed even longer than that. The werewolf as a species predates the printing press, let alone television, movies, and the Internet. Ovid’s Metamorphoses is probably the oldest surviving example of the werewolf in literature. Ovid was contemporary to Jesus, although somewhat older. One version of the “wealthy man turns into a wolf and hunts the forests around his home” was penned by Marie de France, who was born around 1160, in Bisclavret, one of a collection of 12 short narrative poems published in The Lais of Marie de France. More recently, authors Mercedes Lackey (The Fire Rose), Laurell K. Hamilton (the Anita Blake, Vampire Hunter series) and Charlaine Harris (the Southern Vampire series/True Blood) have introduced werewolves and shape-shifters to their preternatural universes.
Finally, the werewolf or wolfman seems to be an archetype of the human psyche. Whether he is a representation of man’s more violent and animalistic nature or just a convenient excuse for barbarous acts and poaching, depends on your perspective.
A better debate might have been the difference between a werewolf and a shape-shifter. What is certain is that The Wolfman did not steal the human turning into a wolf idea from New Moon.