This 2014 video from North America says about itself:
The plumage is the same, but size and a couple of other factors set them apart.
From the Cornell Lab of Ornithology in the USA today:
When two species look similar, it’s natural to assume they’re close relatives. But Hairy and Downy Woodpeckers—two of North America’s best known lookalikes—aren’t closely related at all. Scientists are left wondering how they came to look so similar. One researcher decided to put his ideas to the test.