Sometimes I wonder, is the Nuthatch a woodpecker? Well, they're in the woodpecker section of our bird book.
A web search turns up numerous hits such as "Nuthatches and Woodpeckers", so I suppose they are considered alike but separate.
I say Nuthatch as if the White-breasted is the only one, it's not. While we have had the Rose-breasted, this one here (White-breasted) is the common one in our area. I believe there are others as well, but we've not seen them.