Both the novel The Scarlet Letter and the short story "The Minister's Black Veil" focus on the subject of secret sin, and the letter A and the black veil are both symbols of this.

In the novel, the Reverend Dimmesdale has a secret and illegal affair with Hester Prynne, impregnating her and then failing to confess his share of the responsibility for the relationship. While Hester wears the letter A as a public sign of her adultery, Dimmesdale wears it privately, carving it into his chest. He hides his guilt for seven years and only eventually seems to escape it just moments before he dies, having become a shell of the man he used to be. He acknowledges his own sinfulness but never explicitly confesses it until the last moments of his life; even then, people cannot agree on what he said or did.

In the short story, another Puritan minister, Mr. Hooper, recognizes his own secret sinful nature too, and he comes to believe that everyone actually shares such a sinful nature; however, no one is brave enough to admit it because they believe that everyone else is as sinless as they pretend to be. Mr. Hooper begins to wear a black veil to cover his eyes and nose, forever sealing himself off from the rest of humanity, because he cannot confess his sinfulness directly but feels the only way to live honestly is to wear something which symbolizes his state. For both ministers, their lives are forever altered by the knowledge of their secret sin, and their guilt ultimately keeps them both from engaging fully with anyone else in their lives.

As is characteristic of Nathaniel Hawthorne's style, both the black veil of the Reverend Hooper and the scarlet A of Hester Prynne carry a certain ambiguity to them.  For, as symbols, they have meaning in themselves, but their meaning extends beyond the symbolic object that they wear.  And, in this extended meaning the black veil carries with it much more ambiguity than does the scarlet letter.

Originally, of course, the scarlet letter is the symbolic mark of the adultress; with this mark of sin and scorn, Hester is ostracized from other Puritans and is not allowed to be in certain places or to sow for weddings and baptisms, although she can still flourish the gloves of Governor Bellingham and other dignitaries. Nevertheless there is a certain ambiguity to this scarlet letter as young maidens blush and look away when Hester passes them, feeling their own shame.  But, after Hester accepts her position in the society and  becomes a nurse to the sick and elderly, both the ambiguity and the significance of the letter changes for some as they interpret it to mean "Angel" or "Able" and think more kindly of the now plain woman who quietly works in the community.

On the other hand, the black veil that Mr. Hooper voluntarily dons becomes more sinister in its interpretation as time passes.  Ironically, while essaying to model for his congregation their own hidden weaknesses that they should admit to themselves, many shun him as a the perpetrator of a sin so grievous that he must hide behind a veil.  Others fear that he may know their sins and, so, he disguises his knowledge of their secrets behind the veil.  In addition, the veil itself, being black, becomes a symbol of death, evil, sin, and horror for others; that is, the veil becomes a symbol of symbols. 

Yet, in the wearing of these symbols, both Hester and Mr. Hooper assimilate the symbol into their personalities.  In the final chapter of The Scarlet Letter, Hester returns from England to enter her old home and stoop to retrieve the scarlet letter--albeit it "has not done its office"--replacing it upon her bosom as of old, she indicates that she accepts the letter and the identity that it has placed upon her. In time

...the scarlet letter ceased to be a stigma which attracted the world’s scorn and bitterness, and became a type of something to be sorrowed over, and looked upon with awe, yet with reverence too.

Likewise, even on his deathbed, Mr. Hooper refuses to remove the veil, feeling that this cloth has become part of his spiritual office as minister.But, for Mr. Hooper,the black veil is yet scorned and not revered:

"Why do you tremble at me alone?....Tremble also at each other!  Have men avoided me, and women shown no pity, and children screamed and fled, only for my black veil?  What, but the mystery which it obscurely typifies, has made this piece of crape so awful? When the friend shows his inmost heart to his friend; the lover to his best-beloved; when a man does not vainly shrink from the eye of his Creator, loathsomely treasuring up the secret of his sin; then deem me a monster, for the symbol beneath which I have lived, and die! I look around me, and, lo! on every visage a Black Veil!'’’

As symbols, both the black veil and the scarlet letter possess much import, serving to develop for Hawthorne his themes of secret sin, guilt and innocenc, and doubt and ambiguity in their respective narratives.