Zuko knew his father didn't care who he really was, and his sister only cared if it was useful. His mother and uncle cared, but they weren't here now -- exiled and imprisoned for this ability.
Then there was Mai. He didn't know if Mai knew Zuko, or merely the Prince. He hoped even if she didn't know Zuko, she'd like to find out. He had liked seeing her, not the Dutiful Noble Daughter.
Hence, the leter. Maybe Zuko and Mai could try again, even if the Prince and the Noble's Daughter couldn't.