Leanda de Lisle explains why, in The Spectator:
With his wife, Anne Boleyn, in the Tower, Henry VIII considered every detail of her coming death, poring over plans for the scaffold. As he did so he made a unique decision. Anne, alone among all victims of the Tudors, was to be beheaded with a sword and not the traditional axe. The question that has, until now, remained unanswered is — why?
Historians have suggested that Henry chose the sword because Anne had spent time in France, where the nobility were executed this way, or because it offered a more dignified end. But Henry did not care about Anne’s feelings. Anne was told she was to be beheaded on the morning of 18 May, and then kept waiting until noon before being told she was to die the next day. At the root of Henry’s decision was Henry thinking not about Anne, but about himself.
That last sentence is the key to de Lisle's explanation: she connects Henry's concern with his hurt pride and reputation, disclosed in the trials of his queen and her courtiers with the image of Camelot:
In Malory’s Le Morte d’Arthur, Guinevere was sentenced to death by burning. Henry decided Anne would be beheaded with a sword — the symbol of Camelot, of a rightful king, and of masculinity. Historians argue over whether Anne was really guilty of adultery, and whether Henry or Cromwell was more responsible for her destruction. But the choice of a sword to kill Anne reflects one certain fact: Henry’s overweening vanity and self-righteousness.
What do you think of her explanation?