Kate is not, and can never be, "Princess Catherine".
Her full official title is Her Royal Highness Princess William Arthur Philip Louis, Duchess of Cambridge, Countess of Strathearn, Baroness Carrickfergus - she is all that by virtue of her marriage to William (who's obviously HRH Prince William Arthur Philip Louis, Duke of Cambridge, Earl of Stratheam, Baroness Carrickfergus). It's pretty much the same if she'd married a John Smith; she would assume his 'titles' (surname). .
Kate could only be called "Princess Catherine" if:
a) She were born a Princess (like Princess Anne)
b) She were granted the style and title of a Princess by Her Majesty (like Queen Beatrix of the Netherlands created her daughter-in-law Maxima a Princess in her own right prior to Maxima's marriage to the Heir to the Throne Prince Willem-Alexander)
Neither of the above happened, so she is not entitled to the title of a Princess.
Since Kate isn't a Princess in her own right (she wasn't born one), and had the Queen not granted the new title, Kate would only be known as Princess William of Wales; to avoid the situation, William was created a Duke, so that his wife could be known as a Duchess.
There are 2 kinds of Dukes - Royal and non-Royal ones. Non-Royal Dukes are indeed Nobility, whereas Royal Dukes are most definitely Royal. Traditionally, Royal Dukedoms are granted only to the sons of a Monarch; William is Queen Elizabeth's grandson, however he'll be the King's son one day.
There are currently only 7 Royal Dukedoms:
1. Lancaster - held by queen Elizabeth II (she's the "Duke of Lancaster", not "Duchess" because the title can be held only by the Monarch and has only the masculine form)
2. Cornwall and Rothesay - held by The Prince of Wales (both titles are held by the Heir Apparent to the Throne)
3. Edinburgh - held by Prince Philip
4. York - held by Prince Andrew
5. Cambridge - held by Prince William
6. Gloucester - held by Prince Richard
7. Kent - held by Prince Edward