There are quite a few differences, starting with the eggs. Dragonfly eggs are round, while damselfly eggs are cylindrical. In the larval stage, known as nymphs, damselfly nymphs have three external gills projected back from the tail, while dragonfly nymphs have internal gills.
In the adults, both have two pairs of wings, but with dragonflies, the rear wings are larger, while in damselflies, both sets are the same size. Most dragonflies hold their wings out to the side, while most damselflies fold their wings alongside their bodies.