You would be wise to consider why Hawthorne gave this excellent short story the title that he did. The black veil clearly plays a massively important role, and it is crucial to realise the symbolic significance of the black veil in the story. The black veil represents the secret sins that all of us have and are not honest about. Consider how Mr. Hooper explains himself to his betrothed, Elizabeth:
"If it be a sign of mourning," replied Mr. Hooper, "I, perhaps, like most other mortals, have sorrows dark enough to be typified by a black veil."
Mr. Hooper is therefore wearing the black veil because he has learnt the lesson that others shudder to acknowledge and run away from - that in the heart of every human lies a darkness and sin that we deny and pretend does not exist.
Thus the symbol of the black veil is so potent because it literally separates us from others and even from our Maker as we try to pretend that this darkness is not within us. However, as Mr. Hooper recognises, we can only ever hope to wear this "veil" whilst we are on earth, for when we die, this veil will be stripped away and we will see ourselves for who we really are and others will see us as well. All pretence will be stripped away. This, I would say, would be the most important point of this story. It would be worth your while to consider how the other characters in this story respond to the "truth" that Hooper has discovered and what their response shows about them.