The red nape patch on the male Downy Woodpecker (center photo) is the difference between male and female plumage; female does not have the red patch (lower right photo).
When foraging, generally the male prefers the upper canopy, smaller branches and angled tree limbs while the female favors tree trunks & lower, upright limbs. This may be so the pair does not cross forage when on the same territory. One study showed that when males were removed from their territory, females began to forage in the upper canopy.
Downy Woodpeckers eat mainly insects, especially wood-boring beetles & larvae, also fruit, seeds, sap and suet from feeders. Downy may even take tent caterpillars.