This story conveys the message that what unites all human beings is both our propensity to sin as well as our insistence on concealing our sinful natures from our peers. In this way, then, we can never truly know or be known by another person because we hide behind the figurative veils that we hold up between ourselves and everyone else.
Mr. Hooper's first sermon after beginning to wear the veil was about the subject of "secret sin," the sins that we seek to hide from the world, ourselves, and even God (though such concealment from God is ultimately impossible). Further, his conversation with his fiancee, Elizabeth, hints at this secret sorrow as well, especially when he insists that it is not particular to him but true of all mortals. Finally, in the end, when Mr. Hooper is on his deathbed, he marvels that the veil has been the reason he's spent his life in isolation when he looks at all those around him and says, "'lo! on every face a black veil!'" People have ostracized him alone because of the physical veil he wears, a veil which is only symbolic of the human condition of sinfulness, the condition of all people.