What was Hester's changed in chapter 13?

Hester realized that she and Dimmesdale had each been punished in their own way. She had been publicly humiliated and punished; whereas Dimmesdale had punished himself to the core of his being.

"Hester saw—or seemed to see—that there lay a responsibility upon her in reference to the clergyman, which she owned to no other, nor to the whole world besides. The links that united her to the rest of humankind—links of flowers, or silk, or gold, or whatever the material—had all been broken. Here was the iron link of mutual crime, which neither he nor she could break. Like all other ties, it brought along with it its obligations."

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Whenever anyone was in need, Hester was there to help them. This changed the way people interpreted the badge of shame she wore. Through her good works the symbolism of the letter had changed.

"In such emergencies Hester’s nature showed itself warm and rich—a well-spring of human tenderness, unfailing to every real demand, and inexhaustible by the largest. Her breast, with its badge of shame, was but the softer pillow for the head that needed one. She was self-ordained a Sister of Mercy, or, we may rather say, the world’s heavy hand had so ordained her, when neither the world nor she looked forward to this result. The letter was the symbol of her calling. Such helpfulness was found in her—so much power to do, and power to sympathise—that many people refused to interpret the scarlet A by its original signification. They said that it meant Abel, so strong was Hester Prynne, with a woman’s strength.