This is a great question!
The symbol of the scarlet letter is what, ultimately, seals the deal with Hester and reinstates her back into society as the person whom she found herself to be: A woman whose mistake turns into a never-ending punishment. However, she is the woman whose never-ending punishment has separated her from the rest of a hypocritical society. She is indeed different from them. She has always been. The scarlet letter simply serves as the icon that she will courageously push forward as a badge of challenge to the rest of the settlement: She, Hester, IS the scarlet letter- She is a symbol that should be feared, that should be left alone, and that has a significance of its very own.
The part of the novel in which we find more evident that Hester has come to understand that she and the badge share a common uniqueness, enigma and mystery comes in chapter 5, "Hester and Her Needle", where Hester is witnessed as one of the most talented seamstresses in the settlement. Moreover, her sense of pride drives her to not only decorate beautiful dresses and bonnets; she even takes the time to give her scarlet letter a brand new look.
Her letter is no longer a shabby badge given by the elders as a symbol of shame. Her new scarlet letter is now quite different:
She bore on her breast, in the curiously embroidered letter, a specimen of her delicate and imaginative skill, of which the dames of a court might gladly have availed themselves, to add the richer and more spiritual adornment of human ingenuity to their fabrics of silk and gold.
What does this show us? This shows us that Hester is no longer bothered by having to wear something that only has a bad meaning to others, but not to herself. She is actually proud that she can use that ridiculous letter as a conduit to demonstrate the best she has to give, especially, when the others wait so avidly for her artistic works. She is now beyond the scarlet letter and even more so, beyond the limitations of the settlement. She is a whole woman now, with or without that letter. She has outgrown it.
Therefore, the letter demonstrates that the laws of man cannot transcend the inner power of the human being. Hester demonstrates over and over to the silly men and women of the village who hiss and insult her that regardless of what they think she is still who she is, and that she is quite proud of it. That is much more than any of the settlers could say about themselves.
In the Scarlet Letter Hester Prynne is ostracized for her obvious sin-adultery. She bears the letter, and in doing so, bears what those in the village would tag as her identity. While standing in front of everyone with the A on her chest-she does not weep, she does not falter in her strength. She remains quiet and dignified. She believes this is her only defining quality, this A, the fact that she is an adulteress. She isn't just an adulteress of course, but in embracing her sin-she embraces a part of who she is and begins to live her life not caring what others are thinking of her. She loves her daughter, Pearl, and remains a strong and dignified woman throughout her life.
Arthur was never pegged as the one that committed adultery with her, and he lived in shame in turmoil. He melted away as a strong man by not embracing his sin. It tortured him and weakened him-quite contrary to how it impacted Hester. He did not confess this sin until the end, and it was then that he died in Hester's arms.
Hester became a woman that others would confide it. She was a light to many, even though her sin had made her an outcast before. She remained strong until the end, always living without caring what others had defined her as.