As the Wilsons tour the White House on their first day, they stop to admire the official portrait of President Taft. As Taft had left office only that day, no official portrait of him would as yet have been painted or hung.
When Wilson is elected governor of New Jersey, the movie states that he won all 22 counties. New Jersey has (and had in 1910) only 21 counties.
In the opening of the film, the football game between Princeton and Yale in 1909 is depicted as being played at Princeton with Yale winning 6-0. The game was actually at Yale, and Yale won 17-0. Information from sports-reference.com.
In the establishing shots of Washington, D.C., which are supposed to be on Wilson's inauguration day (March 4, 1913), the trees and grass are green and in full flower, which they would not have been in March, and in particular on the day Wilson was sworn in, which was cold and blustery with snow showers.
When Tumulty comes to the family sitting outside Mrs. Wilson's bedroom during her last illness, he tells President Wilson that "Senator Carter Glass" needs to see him about the Federal Reserve Act legislation. Carter Glass was a sponsor of the bill, but he was in the House of Reppresentatives at the time, not the Senate.
An establishing shot shows the Washington Monument reflected in the west Reflecting Pool, although the latter did not exist when Wilson entered office in 1913.
When Wilson is traveling the country promoting the League of Nations in 1919, cars and trucks from the 1940s are visible outside the train's windows.
American flags are shown with 48 stars prior to the admission of Arizona and New Mexico in January and February 1912.
When Ike Hoover, the White House head usher, tells the Wilsons upon their arrival, "We'll take the elevator up," the character's voice has clearly been looped over by another actor and is not that of the actor portraying Hoover, Roy Roberts.
The servant of Charles Evan Hughes states "the president has gone to bed" when asked if Hughes is available for comment. His title (had he actually won the election) would be president-elect.