Lots of fish besides corydoras have mouth barbels (including most catfish and danios). It is thought that they are an important part of the fish' sense of touch, taste and smell. In some larger catfish, the barbels are strong enough to push aside pieces of substrate as the fish searches for food.
Barbels are also a convenient measure of the fish' overall health. Loss of barbels is usually a very early warning sign of poor water quality and/or a substrate in desperate need of a vacuuming.