In preparation for being executed, Lepke's upper calf was shaven (to provide better electrical conductivity), yet when he is actually executed, the wiring for his leg is placed not on his calf, but just above the ankle.
In the movie, Lepke's trusted partner Jacob "Gurrah" Shapiro is shot to death at Coney Island while trying to protect Lepke from assassins. In real life, Shapiro turned himself in to the authorities, accepted a prison term, and actually outlived Lepke by several years, dying in prison of natural causes in 1947.
Lucky Luciano (Vic Tayback) is shown smugly sitting in the courtroom gallery during Lepke's murder trial. This would not have happened, because this trial occurred in the fall of 1941 and Luciano had been sent to prison in 1936. He was not released until 1946, at which time he was deported to Italy.
For this same reason, Lepke would not have been called out of hiding to attend a meeting at Luciano's house. Lepke went into hiding in 1937 until 1939, and Luciano was in prison this entire time.
Thomas E. Dewey is depicted in the film as the prosecutor of Lepke's murder trial. However, this was a state court case that Dewey was not even involved in as it was not his jurisdiction. The real prosecutors who sent Lepke to the electric chair were Brooklyn D.A. William O'Dwyer and Assistant D.A. Burton Turkus.
As he eats his last meal (while receiving a visit from Robert Kane), Lepke drinks from a water glass. No prison would have permitted such a breach of security, particularly for a condemned man in his final hours, because of the potential use of broken glass as a weapon of suicide or homicide.
When Lepke's leg is being shaved in preparation for the execution, the razor moves across his calf removing the shaving cream but leaving all hair intact: the razor is completely dull.
Throughout the movie, Lepke wears a clearly obvious 1970s hairstyle, including moderate sideburns, even though time frame of the film is the 1920s through the early 1940s. It is even obvious that he is wearing hairspray, a product that didn't become available until the 1950s, and wasn't commonly used by men until the late 1960s at the earliest.
Another example of men's grooming that was much closer to 1975 than to the supposed time frame of the film are the full-brush style mustaches worn by Thomas E. Dewey, as well as by both the Sing Sing executioner and warden.