The most obvious contrast is how they feel about Caesar -- not personally, but in terms of how Caesar's power is growing as the play opens.
Antony is Caesar's right-hand man -- literally. He is always just at Caesar's side. He is so ever-present beside Caesar, that the conspirators discuss murdering him as well as Caesar, since he seems to be so dedicated and attached to him.
Brutus, while seen by Caesar as a loyal compatriot, on the other hand, harbors deep misgivings about the power that Caesar is developing. Brutus is fiercely dedicated to the Roman democratic rule, to the ideals of the Republic. So, when it seems that Caesar would be declared Emperor, Brutus feels that it is his patriotic duty to revolt.
Perhaps the most accessible way to compare their personalities is to compare their funeral orations. Brutus speaks from logic and reason, appealing to the ideology of the people in order to garner their support for his actions. He appeals to their logic and sense of the honorable action, the action that is "best"for the Republic.
Antony, on the other hand, begins with the word "Friends," making it clear from the beginning that it is not their responsibility as citizens to which he will appeal, but their emotions, their feelings. Suffice it to say, Antony's speech over Caesar's dead body creates an uprising of the citizens that drives the conspirators out of Rome.
Overall, Antony is the sort of man who operates based upon his emotions and is able to garner support based upon this mode of operation. Brutus, on the other hand, operates based upon logic and sacrifice for the good of the Republic, something that does not reach the common man in the same passionate way that Antony can.