A Furry (sometimes called just a "Fur") is a member of the "Furry Fandom", people who have an interest in anthropomorphic animals in art, fiction and media.
While the Furry community is large and diverse, and while there is some amount of cross-over between the "Were" and "Furry" communities, Furry generally indicates a lifestyle choice or interest in anthro animals, rather than that a person actually considers themselves to be an animal.
There are furry lifestylers who do feel a strong and profound link to their animal nature, and self-identify as being that type of animal at heart.
Being a were and being a furry are two different things. Being one does not necessarily make you the other, but being one does not necessarily prevent you from being the other, either.