Actually you've got it backwards.
A Royal Dukedom (a Dukedom held by a Prince) is a higher title than simply being a Prince. If you're a Duke then you're just a Duke, and if you're a Prince you're just a Prince, but if you're a Royal Duke you're both a Prince and a Duke.
William's full title is His Royal Highness Prince William Arthur Philip Louis, Duke of Cambridge, Earl of Strathearn, Baron Carrickfergus, Royal Knight Companion of the Most Noble Order of the Garter, Knight of the Most Ancient and Most Noble Order of the Thistle. This is often shortened to simply HRH Prince William, Duke of Cambridge, KG, KT. To shorten it even more, sometimes he gets called HRH Prince William (because that's what people knew him as for so long), and sometimes he gets called HRH the Duke of Cambridge, which is the proper way to refer to him.
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