Edwin M. Stanton had a long, gray beard at the time of the assassination and thereafter. He also bore a heavy build. However, some historical films are known for having 'non-impersonations' of a figure's appearance, they don't always have to be accurate with the appearance.
Mary Simms was not still working for Dr. Samuel Mudd at the time he notoriously treated John Wilkes Booth. She had quit a year earlier in 1864.
Edwin Stanton's role in the investigation/manhunt is quite exaggerated. He didn't play that large of a part in it as depicted in the series.