Summarize:

Both species have short, sharp calls.  However, if you listen carefully, you can detect slight differences in the calls.  The Downy Woodpecker's pik call is a rapid string of high-pitched notes that descends in pitch toward the end.  Their call typically lasts about two seconds.  Hairy Woodpeckers, however, make a similar short sharp note, described as more of a peek sound.  It is slightly lower in pitch and does not descend at the end as the Downy's call does.  Hairy Woodpeckers also have a rattling or whinnying call. The males and females of both species use their bills to drum on trees as a means of communication.  However, Downy's drum is slightly slower than that of a Hairy, about 17 beats per second as opposed to the Hairy's 25 beats per second. Both species primarily eat insects, along with some fruit and seeds. But there are some differences in how the two species feed that may help you identify them.  Because of their smaller size, Downy Woodpeckers can eat foods that larger woodpeckers can't access, like insects in the stems of larger weeds.  Hairy Woodpeckers never feed on weeds. Hairy woodpeckers like to drink the sweet sap of trees.  They may even peck into sugar cane to drink the sugary juice within. This can be tricky because both are found in woodlands and may often flock together.  However, the Downy Woodpecker favors smaller branches while the Hairy spends more time on large branches or trunks.
Listen for differences in their calls. Listen for different drum sounds. Pay attention to different eating habits. Consider differences in location.