When you weep, you cry. If you're very empathetic, you might start to weep whenever you see other people weep.
To shed tears from sadness is to weep. Some people weep rarely, only when they've suffered a true loss or tragedy. Other sensitive souls weep at songs, movies, books, and even sweet stories or pictures. Weep comes from the Old English word wepan, "shed tears or mourn over," which has a Proto-Indo-European root meaning "to cry or scream."