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.
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.
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.
At 2:11:00 of the movie, Wilson (Alexander Knox) signs the Treaty of Versailles, a stage prop which contains a number of errors: (1) In the real document, the real Wilson's signature appears on a blank page following the end of Article 440, not immediately beneath the article. (2) In the real document, the word "transmitted" isn't hyphenated across a line break. (3) In the stage prop, the word "transmitted" is incorrectly hyphenated "tran-smitted" - it should be "trans-mitted".
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.
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.