At the battle of Cannae, Hannibal states that the Romans had the river at their backs. Actually the river was on the Roman's right flank and to Hannibal's left.
The elephants shown in the movie are Asian elephants, as is recognizable from the twin-domed head with an indent running up the center of their head. This is not the case with African elephants.
Carthaginian frescoes and coins minted by whoever controlled North Africa at various times show very small elephants, perhaps 2.5 metres (8 ft 2 in) at the shoulder, with the large ears and concave back typical of modern African elephants. The North African elephant was smaller than the modern African bush elephant (Loxodonta africana africana), probably similar in size to the modern African forest elephant (Loxodonta cyclotis).
Maybe the Asian species was used, because it is typically smaller in size than the African species.
Towards the start of the film there is a panoramic shot of Rome. This shot includes many familiar buildings such as the Colosseum, which was not built until some 300 years after the events of the film.
Crossing the Alps there are many shots of men climbing steep inclines, using ropes, hand over hand but no explanation of how the elephants or horses get up the same steep inclines or where their spears and shields go that they carry while on the flat.