U.S. v. Nixon cites the case of U.S. v. Burr, in which Aaron Burr, who was most famous for smoking Lin-Manuel Miranda—uh, we mean Founding Father Alexander Hamilton—in a duel while serving as vice president, was being tried for treason.
Even though dueling was illegal in New York at the time, that's not why Burr was in court.
After his VP gig, Burr moved to the Western frontier. In 1807, he was accused of planning to lead an unauthorized invasion of Mexico, and of attempting to sever part of the United States into an independent country which he would rule. During the trial, Burr wanted the court to subpoena President Thomas Jefferson, to see what evidence would be used against him. The evidence was Jefferson's private letters, which TJ claimed would endanger public safety if revealed.
Chief Justice John Marshall decided that the courts, not the president, would decide that. They decided that Jefferson had to hand over the letters, and Jefferson complied, and handed over the requested documents. This is the first time that "executive privilege" would be invoked by a president.
Marbury v. Madison, also cited in the U.S. v. Nixon decision, was the first time that the Supreme Court ruled that a law could be unconstitutional. It formally established the right of judicial review, i.e., that laws can be reviewed by the judicial branch to evaluate their constitutionality.
In Marbury, the Court ruled that President Thomas Jefferson was wrong in preventing William Marbury, who was appointed by his ex-bestie John Adams, from taking his appointed office as justice of the peace. Unfortunately for Marbury, the Supreme Court also claimed the case that allowed Marbury to bring his suit was unconstitutional, and couldn't force Jefferson to allow Marbury to take his appointment.
This is one of the first cases that addressed the separation of powers among the branches of government. The reason that Marbury v. Madison was referenced in the U.S. v. Nixon was to reaffirm that the Supreme Court has the final word in determining constitutional questions, and to affirm that no man, not even the president, was above the law.
While the cases are the same, and have to do with the jurisdiction of the courts, their outcomes were different. While the Supreme Court didn't force Jefferson to seat Marbury, it did force Nixon to surrender his tapes to the Watergate Committee.
Clinton v. Jones mentions U.S. v. Nixon in its argument. President Bill Clinton was being sued by former Arkansas state employee Paula Jones for sexual harassment while Clinton was governor of Arkansas. The big question in Clinton v. Jones was: based on the separation of powers argument, can a sitting president be sued for actions that were committed before he was president? U.S. v. Nixon set precedent for this case, arguing that the president is not above the law, and is subject to the courts under special circumstances (like this one). The ruling in Clinton v. Jones would confirm that the president has no immunity from civil litigation for acts done before taking office.
The Clinton case was a civil, not a criminal one, but the outcome was the same: presidents are not above the law.
Here we go again with Clinton.
During the Monica Lewinsky investigation (let this SNL clip fill you in), President Clinton claimed executive privilege in an attempt to limit the questioning of two of his aides during the investigation. The Supreme Court decided that President Clinton could not use executive privilege to shield his top aides from questioning. As in U.S. v. Nixon, the Supreme Court decided that the interest of the investigation outweighed the executive privilege of the president.