A good comparison of a Downy Woodpecker and Hairy Woodpecker on my suet log today.

The smaller bird on the left is the Downy, and the larger is the Hairy.
A good field mark if you see a lone bird, is to look at the length of the beak compared to the depth of the head.
The beak of the Hairy is approximately as long as the depth of the head.
A Downy's beak is about half as long as the head depth.