In The Scarlet Letter, why does Hester refuse to stop wearing the letter?

When the Reverend Mr. Wilson questions Hester to get her to name her child's father, Hester repeatedly refuses. He tells her that revealing this name and her own repentance "'may avail to take the scarlet letter'" from her breast. She replies, "never, [...]. It is too deeply branded. Ye cannot take it off. And would that I might endure his agony as well as mine!" Hester seems to imply that she cannot stop wearing the letter because it isn't just a piece of cloth that she pins to her clothes; it seems somehow more deeply impressed into her very body. She feels that it is "branded," which makes it seem as though it is something she can never, ever remove because it is a part of her now.

In the end, as an old woman, Hester returns to Boston of her own volition and begins to wear the letter once again. "Never afterwards did it quit her bosom." It also seems that Hester has some admirer, a man who sends her letters and gifts that pepper her home. However, she seems to return, and to put the letter back on, because it links her to Dimmesdale, the man she evidently still loves. The letter becomes "something to be sorrowed over, and looked upon with awe, yet with reverence too."

Hester is required by the Purtain culture to wear the scarlet letter as a reminder to herself and everyone else of her adultery. However, as time passes, the letter begins to mean more than just a reminder of her troubles; Hawthorne alludes to this in several passages ("A" meaning "Able" or "Angel") as Hester, although living on the fringe of the society (both figuratively and literally), becomes that society's helper. The red letter ceases to be a mark of shame but becomes a badge of honor.

What is the relationship between the scarlet letter and Hester's identity in The Scarlet Letter, and why does she refuse to stop wearing it?

In Chapter XX, Nathaniel Hawthorne writes,

No man, for any considerable period, can wear one face to himself, and another to the multitude, without finally getting bewildered as to which may be the true.

After wearing the scarlet letter for so many years, Hester's self-conception alters.  This letter, whose wearing has caused her beauty to fade and her passion to die, has become her identity.  There is little difference between the symbol and the woman; it is not a cover for her identity, but, rather her actual identity.  This is why Hester returns after having lived in England with her daughter Pearl and resumes the wearing on the old, faded letter that lies in the doorway of her old cottage's threshold (Chapter XXIV):

On the threshold she paused--turned partly round--for, perchance, the idea of entering all alone, all so changed, the home of so intense a former life, was more dreary and desolate than even she could bear.  But her hesitation was only for an instant, though long enough to display a scarlet letter on her breast.