I thought this might be entertaining: I'm wondering how you think the world might be different if some very crucial events did not happen or happened differently. Since some events are going to be past others, I'm going to say pick one and roll with it from there. (These examples will be some obscure history and some common history in the ancient era) Feel free to come up with your own, these are just a few of mine.
Alexander the Great is killed in Anatolia (Turkey) while fighting the Persians at the Battle of Issus, suffering a terrible defeat.
Hannibal Barca succeeds in his Italian Campaign, besieging and even conquering the city of Rome; the Carthaginian Senate follows through with his demands for reinforcements instead of bickering until Scipio Africanus lands.
Julius Caesar suffers an incredible defeat at the Battle of Alesia, Vercingterox kills Julius Caesar by running him down while Ceasar and his legions are retreating.
Julius Caesar does not burn the Library of Alexandria during the siege of Alexandria (or indirectly burn).
After Caesar's assassination, Brutus and Cassius were victorious over Mark Antony and Octavian at the Battle of Philippi, thus ending the Roman Empire before it began. (Brutus and Cassius were the liberatores, fighting for the Republic)
During the Year of Four Emperors, Vespasian fails to take over Rome (I have no idea how this might happen, it was a pretty crushing defeat for the defenders of Rome)
Constantine the Great dies somehow before beginning his Christian conversion of the empire and before he accidentally creates the Great Schism.
I'll post my choice and outcome after a few responses have been made