ANOTHEOLDGIT from the official royal website,
"The Royal Family name of Windsor was confirmed by The Queen after her accession in 1952. However, in 1960, The Queen and The Duke of Edinburgh decided that they would like their own direct descendants to be distinguished from the rest of the Royal Family (without changing the name of the Royal House), as Windsor is the surname used by all the male and unmarried female descendants of George V.
It was therefore declared in the Privy Council that The Queen's descendants, other than those with the style of Royal Highness and the title of Prince/Princess, or female descendants who marry, would carry the name of Mountbatten-Windsor."
Since William is a Prince and an HRH, he does not have the Mountbatten-Windsor surname.