The Scots claimed the bragging rights with a 14-5 win in Pool C, but they also failed to make the last eight.
England's 22-7 win over pool winners Portugal was too little too late, while Scotland finished with a 12-12 draw with Samoa.
There was better news for Wales, though, as they progressed to the quarters thanks to a 19-14 victory over Australia.
Their 31-0 loss to South Africa in the final game of the day could not stop them finishing top of Pool A.
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