Railing against how cold it will be outside on the dais during the inauguration, Abbey Bartlett asks "Who in his right mind decided that January would be the right time to hold an outdoor ceremony north of the equator?" President Bartlett replies "Jefferson, Adams, Franklin". President Bartlett, being a fan of historical minutiae, would know better. Inauguration Day, was on March 4 from 1793 until 1933. It was changed to January 20th by the 20th amendment.
When introducing the president-elect and Mrs. Santos at the start of the inauguration ceremony, the announcer correctly pronounces Helen Santos' last name correctly ("San" sounding like "San Diego"), but while then introducing Matthew Vincente Santos, he pronounces the last name "SAHN-tos".
The correct pronunciation was pointed out to Leo McGarry by Annabeth Schott while on the campaign trail. The person whose job it is to introduce the president-elect at his own inauguration would never pronounce the name incorrectly.
When Kate Harper and President Josiah 'Jed' Bartlet are discussing the train derailment between Haverhill (pronounced "Haive-rill"), Mass. and Exeter, N.H., Kate pronounces "Haverhill" like halve-er-hill, which Bartlet, being from New Hampshire, would have at least grimaced at, if not corrected her outright.
When Ronna comes in to the oval office near the end to remind President Santos that the first lady is expecting him in 15 minutes, the time on the grandfather clock changes from approx 5:34 to 5:47. In a slightly later shot in the same scene you can see (albeit blurry) that the clock is back at approx 5:34 again.
At the inauguration, Matt and Helen Santos are waiting to be announced. They kiss, his name is announced and he walks out to the top of the stairs and waves but Helen is already at the bottom of the stairs waiting for him to walk down.
When ex-President Bartlet is returning to Manchester, NH in Air Force One (from Washington DC), the plane is shown over an open expanse of water. Since Air Force One has priority clearance, the direct flight from Washington to Manchester would not be over any open water, such as the Atlantic Ocean or even Chesapeake Bay.
As the new President approaches the Marine honor guard as they are on their way to the inaugural podium, the Sergeant of the Guard gives the command "Order - Arms" as the honor guard presents arms. The command should have been "Present - Arms".
When Abbey asks Jed who came up with the idea of having the Inauguration on January 20, Bartlet replies "Adams, Jefferson, Franklin". It has already been mentioned that Inauguration Day was March 4 until the 20th Amendment. In addition, neither Adams nor Jefferson were at the Constitutional Convention (Adams was the ambassador to the British, and Jefferson to the French).
On January 20, the sun sets at about 5:15 pm. It would not be possible for sunlight to be streaming in the windows of the Oval Office or of the Bartlet Airplane at about 6:00 pm on the day of the Inauguration.
Both the justice administering the presidential office and Matt Santos get the oath of office wrong. In the phrase, "I will faithfully execute the Office of President of the United States," they both incorrectly add a definite article before the word 'president' (about 12 min. before the end).
After CJ exits the gate, the north lawn of the White House is shown without any setup for the Inaugural Parade. The President's private booth, or any of the bleachers lining Pennsylvania Ave., would've still been in place.