Robert Todd Lincoln is taken with Lucy and asks her to escort him to the theatre. In reality, Robert Todd Lincoln was already courting his future wife, Mary Harlan, at the time of his father's assassination. Both of Robert Lincoln's parents knew of, encouraged, and helped with the courtship, which had been going on since the beginning of 1865. In fact, Robert and Mary's marriage was delayed due to the assassination and the subsequent mourning period. While Abraham Lincoln may certainly have given a polite greeting to a woman with Robert, the President and/or Mrs. Lincoln would certainly have asked why Mary was not with him.
The Union Army uniforms given to Rufus and Wyatt have bugle cap devices indicating Infantry and Rufus introduces himself to the soldiers as Infantry, but their trouser seam stripes and respective sergeant and corporal rank stripes are red which indicate Artillery. (Infantry trouser seam stripes and NCO stripes are light blue.) A reenactment group would not have provided such an obvious mismatch in insignia, and it would have been an immediate giveaway that they were impostors well before the soldiers approached Rufus.
They show Lincoln dying the same night he was shot. In reality he lived until the next day.
At the beginning of the episode, Ward Hill Lamon says the President (Abraham Lincoln) is next door at Ford's Theatre, watching "Our American Cousin". In reality, Lamon wasn't in Washington, D.C. at the time, as he had been sent to Richmond, VA by Lincoln.
Ford's Theatre is on the west side of 10th St. NW in Washington. The view shown, therefore, would be looking north, away from what would become the Mall. That and the fact that the Capitol is nowhere near in a direct line of sight from 10th St. in any direction means that, in reality, the Capitol would not be visible at all.
There were dozens of experienced politicians who could have stepped in if the full assassination plot had succeeded. Flynn would have known this.
After they arrive in Washington on the eve of Lincoln's assassination (April 1865), Lucy says that slavery has ended. However, slavery wouldn't end until the adoption of the 13th Amendment, which didn't happen until December 1865.