Nixon tried to cover up the fact that he ordered the 1972 Watergate break-in. The actual crime is called "obstruction of justice."
Clinton was accused of lying under oath during a lawsuit, which is perjury. He may also have been guilty of obstruction of justice.
How is the President involved?
The president ordered the break-in and then used the powers of his office to cover up the conspiracy. The President claimed "executive privilege" in that he refused to release information on the cover-up to Congress.
The President was involved in a civil lawsuit involving sexual harassment. During the deposition, he gave false and misleading responses to the questions put to him by the special prosecutor.
How is the Congress involved?
The House of Representatives had begun impeachment hearings to determine if a vote by the full house was in order. Once the hearings had begun, Nixon resigned his office, the first president ever to do so.
The House of Representatives did bring articles of impeachment against President Clinton after the House Judiciary Committee found that there was enough evidence to impeach the president. The Senate
found the President not guilty, lacking the needed two thirds vote to convict and remove from office.
How is the Supreme Court involved?
The Supreme Court ruled against the President in US v Nixon ruled "Neither the doctrine of separation of powers, nor the need for confidentiality can sustain an absolute, unqualified Presidential privilege of immunity from judicial process."
The Supreme Court played very little role in the Lewinsky scandal other than the constitutional role played by the Chief Justice in the impeachment trial.