Here are two images side by side for comparison. On the left is the Hairy Woodpecker and on the right is the Downy Woodpecker. Both have similar white and black stripes on the head, and the red on the Hairy is just an indication that this individual is a male. Male Downy Woodpeckers also sport the red patch. However the feature to note is the much longer bill of the Hairy. The bill is nearly as long as the head itself whereas the Downy's is much shorter and also covered with much more downy fluff at the base of the bill.
Downys have black markings on the outside tail feathers, if you are lucky to catch a glimpse of them, which are lacking on the Hairy. Expand the images to take a better look.
Can you tell who is who in these two pictures?
Below is a series of images I have taken over the years of the different species of woodpecker that have appeared in my yard. I was so pleased late last season to have a dozen or more Northern Flickers in a tree feasting on the fruits it produced.