Historically speaking, the answer is very short: because there was conflict between Antony and Octavius. Initially they were sort of partners in a triumvirate, with Marcus Lepidus, to govern the Roman Empire. Here is how the Britannica Concise Encyclopedia describes a triumvirate:
In ancient Rome, usually a board of three officials who assisted higher magistrates in judicial functions, oversaw festival banquets, or ran the mint. The First Triumvirate (60 ) of Pompey, Julius Caesar, and Crassus was an informal group of three strong leaders with no sanctioned powers. The Second Triumvirate (43 ), consisting of Mark Antony, Lepidus, and Octavian (later Augustus)—formally tresviri rei publicae constituendae (“triumvirate for organizing the state”)—held absolute dictatorial power.
Antony even married Octavius's sister. However, Antony was in a relationship with Cleopatra and spent most of his time in the east with her. Eventually each man wanted to be the only one in control and conflict arose. Cleopatra financed the building of a navy for Antony, but they were defeated by Octavius at the battle of Actium. Rather than be taken to Rome as a prisoner, Antony killed himself by "falling on his sword," and Cleopatra famously died by letting an asp bite her.
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Per history, the conflict shown by Shakespeare throughout the plot is a foreshadowing of the real conflict that occurred between the two men after the assassination. Although Antony and Octavius will ally for a time, working together to defeat the army of Brutus and Cassius, they will eventually raise armies against each other. Antony will be defeated by Octavius in 33 B.C. and Octavius will shortly thereafter become Emperor Augustus.