To begin, both men are of elevated status, Hamlet is a prince and Antony ruler of one third of the evolving Roman Empire.
Each man, it would seem, would have been much happier without their elevated staus. Hamlet seems content to stay at Wittenburg as a professional student and Anony would have more than happy to wile away the time in Cleopatra's arms gliding down the river Cydnus.
Both were private men thrust into a public world, and not just any public world, the world of politics, which seems to be just as cut throat then as it is now.
They differ as to how they got to where they are at the beginning of each play.
Hamlet is charged by the ghost of his father to revenge his murder by his brother, Hamlet's uncle/step-father. It is a matter of family honor. Hamlet succeeds in his goal but the cost in lives is great.
Antony has chosen the world of politics but longs to retire and live happily in Egypt with Cleopatra. His marriage to Octavia is political. His great love is the exotic queen who brings his downfall.
In both cases there is no other possible ending for either man and that is tragedy for each of these basically good men.
By definition, the tragic hero falls due to a flaw in his or her character. Hamlet's tragic flaw is his indecision. He had ample time to get his revenge on Claudius, his uncle, who murdered Hamlet's own father. Hamlet, however, chose not to act until he was sure that Claudius was guilty. Once Hamlet is convinced, it is way too late to do anything without multiple victims. Because of Hamlet's indecision, Laertes, Gertrude, Polonius, Ophelia, and Hamlet himself all die.
Unlike Hamlet, Antony's main character flaw is the beauty of the lovely Cleopatra. He is completely taken by her, manipulated by her, and by the time he realizes his error, he doesn't even care that he is too far gone.