Royals use TITLE NAMES more than surnames. Title names identify which branch of the family the member is from. Charles is Prince of Wales and his boys used Wales in school and the military. William was recently given his own peerage title. William's branch of the family now use his title name, Cambridge, from his title, Duke of Cambridge, to differentiate themselves from the others.
The family surname Mountbatten-Windsor or Windsor is used rarely by those who have titles.
EDIT: The British Royal Family LEGALLY adopted a surname in 1917!
http://www.royal.gov.uk/ThecurrentRoyalFamily/TheRoyalFamilyname/Overview.aspx
...In 1917, there was a radical change, when George V specifically adopted Windsor, not only as the name of the 'House' or dynasty, but also as the surname of his family. ..At a meeting of the Privy Council on 17 July 1917, George V declared that 'all descendants in the male line of Queen Victoria, who are subjects of these realms, other than female descendants who marry or who have married, shall bear the name of Windsor'. ..
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.