There is a brief passage that involves the discussion of the black woodsman's identity, but he is never specifically named as the Devil; in fact, in a later exchange between them, Tom says he'll send people "to the Devil" as though he were unaware that the Devil is the woodsman - although I think in this case Tom is just using a grandiose figure of speech and is perfectly aware of who he's dealing with.
The woodsman is described as being literally black, as though painted black, with a soot-smeared face that one would expect to find on a blacksmith. His hair is black, coarse and unruly, and he has red eyes. He wears a "half-Indian garb", meaning that his clothes are some mix of a woodsman's typical attire and Native American elements, with a red sash or belt, and an axe. He has a growling way of speaking and can be somewhat polite but also appear scatterbrained or preoccupied.
He is also described, or attributed, more explicitly supernatural abilities, such as burning his fingerprint into Tom's forehead, and appearing to sink into the earth.
Other names he is given include the "Wild Huntsman", "Black Woodsman", and "Black Miner", but Tom identifies him as "Old Scratch". Scratch appears to consider all of these names of equal value and doesn't really attach special significance to them.