Executive privilege is an implied power. Therefore it is not mentioned in the Constitution. Although "executive power" states that the President can withhold information from other government branches due to separation of powers. In the case United States V. Nixon it does not apply because it does not allow justice to be made in the case.
In the Case of United States V. Nixon its significance is that a U.S. President has never been tried of a crime and taken to the supreme court before.