President Bartlet's second inauguration is portrayed as happening on a Sunday. When January 20th happens on a Sunday, while the President is still sworn in on that day, the actual inauguration always takes place the next day, on Monday, with the oath repeated at the full public ceremony. However, see the Trivia for why this might not be a goof on The West Wing.
When CJ is talking to Toby & Will at the Capitol, she says (referring to the music in the background) "That's the U.S. Marine Corps Band right there, The Commandant's Own." The U.S. Marine Band is called "The President's Own" whereas "The Commandant's Own" is the U.S. Marine Corps Drum and Bugle Corps. They are two separate bands, and it's more likely that "The President's Own" is the band that would be playing at the inauguration. Moreover, the Commandant's Own is the most elite among the Marine Corps Bands, all of which are comprised of enlisted personnel and commanded by career officers. The President's Own is called, simply, The Marine Band. It is an auditioned group that attracts some of the nation's most gifted wind and percussion players. While members of the President's Own are technically marines, they are not subject to combat deployment. Their only assignment is guaranteed to be at the President's call to play their instruments. Members of The Commandant's Own and all other musicians in the US Marine Corps Band program are fully trained and ready to lay down their instruments, pick up rifles, and fulfill any assignment necessary at any time.
The President (Bartlet) is shown riding on the left side of the car. In real life, the President always rides on the right (behind the front passenger) side of the car. That way both the driver and the secret service agent in the front can see him at all times.
On Matt Santos' inauguration his wife asks why the President has to be sworn in on January 20 when it is so cold. Santos responds "the Founders chose it ". He is incorrect. In fact the Founders chose March 20. The date was only moved to January many years later.
CJ mentions Zaire being in talks with Equatorial Kundu. The "Republic of Zaire" was named back into the "Democratic Republic of the Congo" in 1997, after dictator Mobutu fled the country.
The Ronald Reagan Building and International Trade Center garage was used when President Bartlet supposedly arrived at the U.S. Capitol for his swearing-in. The sign is partially blocked by the police officers during Bartlet's walk from the limo.