Brutus and Antony really have two entirely different objectives. There is a lot of turmoil in the streets, and Brutus is mainly interested in calming everybody down. That is why he appeals to reason. For example:
Who is here so base that would be a bondman? If any, speak, for him have I offended. Who is here so rude that would not be a Roman? If any, speak, for him have I offended. Who is here so vile that will not love his country? If any, speak, for him have I offended.
Brutus's speech is calm, well-ordered, rational, and unemotional. Antony, on the other hand, is not interested in calming things down but in stirring things up. He announces this in a soliloquy even before he makes his funeral address.
Over thy wounds now do I prophesy
Which like dumb mouths do ope their ruby lips
To beg the voice and utterance of my tongue,
A curse shall light upon the limbs of men;
Domestic fury and fierce civil strife
Shall cumber all the parts of Italy;
Blood and destruction shall be so in use,
And dreadful objects so familiar,
That mothers shall but smile when they behold
Their infants quarter'd with the hands of war;
All pity choked with custom of fell deeds,
And Caesar's spirit ranging for revenge,
With Ate by his side come hot from hell,
Shall in these confines with a monarch's voice
Cry “Havoc!” and let slip the dogs of war,
That this foul deed shall smell above the earth
With carrion men, groaning for burial.
Antony wants "domestic fury," "blood and destruction," "revenge," "Havoc!" Naturally his speech will be entirely different from that of Brutus. Antony will appeal to his audience's emotions in every way he can. This will include showing them Caesar's bloody cloak and then removing the ruined garment and showing them the bloody body. He will remind the citizens of Caesar's many conquests and of how they all loved him. Antony's own sincere feelings of grief and outrage will be communicated to them through his tears and his voice. Antony is reckless. This has always been one of his noteworthy characteristics. He has everything to gain here and nothing to lose. He is all alone against the whole organized group of conspirators. He keeps Caesar's will as his ace-in-the-hole and uses it near the end of his speech to show how much Caesar loved the common people. It must seem pretty obvious to the greedy plebeians that if the conspirators murdered Julius Caesar, they are hardly likely to honor the terms of his will which calls for giving each of them seventy-five drachmas. When the citizens finally break out into a murderous riot, Antony has achieved his purpose. He says:
Now let it work. Mischief, thou art afoot,
Take thou what course thou wilt.
Brutus wanted a peaceful, orderly culmination to Caesar's assassination. Antony, motivated both by his sharp mind and his turbulent emotions, wanted chaos. He probably had no idea what the outcome would be, but he must have sensed, correctly, that disorder would be to his advantage at that time.
This is the crux of the matter. All seemed to be going well for Brutus and the conspirators, but Antony's brilliant speech undermined their work. Antony did this in three ways.
First, from a oratorical point of view he used a lot of repetition. This allowed the basic point to settle in the minds of the hearers. What made this particularly powerful was that Anthony knew that the crowd was fickle. In other words, they could be swayed by words, and he did just that.
Second, Antony's speech was clever in that it showed deference to the conspirators, but at the same time it undermined them. Read these lines:
He was my friend, faithful and just to me. But Brutus says he was ambitious, And Brutus is an honourable man. . . . When that the poor have cried, Caesar hath wept. . . . Yet Brutus says he was ambitious, And Brutus is an honourable man. . . . I thrice presented him a kingly crown, Which he did thrice refuse. Was this ambition? Yet Brutus says he was ambitious, And sure he is an honourable man. (III.ii.82–96)
By juxtaposing the refrain, Brutus is an honorable man, with the goodness of Caesar, Antony calls Brutus's honor into question. By the end it is clear that the really honorable man is Caesar, who has been unjustly killed. And the refrain that Brutus is an honorable man becomes sarcastic.
Finally, when Antony reads the will of Caesar, he clinches it. He wins over the crowd for good, as all can see the generosity of Caesar.
Thank you! (: