Queen Elizabeth II's likeness faces to the right on coins and to the left on stamps due to traditions within the Royal Mint and Royal Mail. Ever since the reign of Charles II, portraits of British monarchs on coinage alternate the direction they face based upon their immediate predecessors. The Royal Mail always shows monarchs facing to the left on standard postage stamps.
The one exception to the coinage rule was during the brief reign of Edward VIII, who abdicated less than a year after assuming the throne. He preferred his likeness going to the left even though tradition dictated his portrait should face to the right. Designs for proposed coins show Edward VIII facing left even though no coins were minted. Edward VIII's successor, George VI, restored the tradition by facing his likeness to the right.
The Royal Mail has printed stamps with the monarch facing left ever since 1840 when the first stamps were made. The original "Penny Black" shows Queen Victoria's bust facing left. Standard stamps up to July 2014 have the monarch facing left, including the 1967 design used during most of Elizabeth II's reign. The stamp rule doesn't apply for commemorative issues that have shown full portraits of monarchs and busts facing to the right.