Commodus is referred to in the movie as Marcus Commodus. Never in his life did he ever have that name, nor was he ever known by that name.
The film shows a maniacal Commodus attempting to wipe out the Christians, but the real Commodus did not pursue his father's prohibitions against the Christians. He relaxed persecutions after minor efforts early in his reign. It is even believed that the reign of Commodus inaugurated a period of numerous conversions to Christianity.
In the film Marcus Aurelius' dying wish is that Commodus keep peace. Instead Commodus begins attacking villages. In real life it was actually the opposite. The real Commodus ended the fighting when he became emperor. His reign was a time of peace. He did not pursue his father's wars and he organized a truce with the Danube armies Rome was fighting.