In his narration, Claudius says that Livia was Augustus' second wife. In reality, she was his third. His first wife was Clodia Pulchra while his second wife was Scribonia, the mother of his only biological child Julia.
When Drusus is dying, the soldier tell Tiberius that Drusus has "gangrene". Gangrene was not discovered and named until 1883 by a French physician.
Just after Marcellus dies, Livia speaks of Tiberius having a wife, but in reality, he was not married until four years after the death of Marcellus.
Marcus Agrippa appears much older than Augustus, and this is because the actor John Paul who plays him was about fifteen year older than Brian Blessed, who portrays the Emperor. Agrippa complains that he himself is an old man, and yet both he and Augustus were born in the same year - 63 AD, so at the time both were only in their late thirties.