By the early 18th century, the British use of the Julian calendar was at odds with much of the Continent, which used the Georgian. In writings of the period, very often dates were dual: O.S. and N.S. for Old Style and New Style. Washington's birthday, then, would have been February 22, 1732 N.S. and February 11, 1731 O.S. Both were correct.
In 1751, Parliament finally got around to aligning Britain with the Gregorian calendar, as well as moving New Year's Day to January 1, effective in 1752. This site isn't the place for a detailed discussion of the history of the Julian and Gregorian calendars, but much more can be found about the change in British dating systems here and (amazing what's on the Internet) the text of the act of Parliament that made the change.
Some sources say Washington continued to celebrate February 11, others that he made the switch to be mathematically correct. In any case, New Style has won out in the long run, and school kids learn (or ought to) that Washington's birthday is February 22. But on the 11th, there should be a nod to the original birthdate of the Father of Our Country.