Brutus vastly underestimates Antony and over estimates his own abilities. Antony is no man's fool and says what he knows that he must in order to survive. His mind is already at work as he speaks with Brutus and Cassius. He understands these men and their intentions. He plays up to Brutus's honor and integrity since he knows Brutus is in charge and has the final word on the subject.
Antony is an opportunist. He see his chance and seizes it. His soliloquy at the end of the scene tells the truth. He asks forgiveness for the shame of being "so meek and gentle" with Caesar's killers. He predicts a war so horrific that mother would smile to see their children quartered by war. He releases the dogs of war when he turns the mob against Brutus and the rest of the conspirators. He doesn't seem to care who gets hurt and how many innocent people die. (i.e Cinna the Poet) He will have his revenge, plus this is his chance at power and he becomes a member of the Triumvirate.
The bottom line is that Brutus being a trustworthy man trusts Antony where as Cassius does not. Cassius proves correct in his suspicions. Cassius and Antony are politicians. Brutus is a statesman. Cassius and Antony recognise each other and their motivations. Brutus is too naive. He belief in the innate goodness of people is part of his downfall.
As is so often the case in history, a man's virtue turns into his nemesis. Such is the case for Brutus. Because he himself is honest, trustworthy, and equitable, he, therefore, projects this nobility onto others. Brutus would not manipulate people over the dead body of his ruler and friend, so he can not imagine how Marcus Antony can.
However, Cassius is not nearly so noble in spirit. Having been deceptive himself in the "seduction scene" of Act II, implying that Ceasar is a Collossus under whose legs the petty men walk and peep about, Cassius convinces Brutus that the ruler must be killed. After this occurrence, Brutus seems to disregard Cassius's advice, not tusting him as he perceives his evil.
Perhaps because Brutus perceives the injustice of Ceasar, and perhaps because of his desire to be perceived as a just and noble man, he fails to apprehend the danger of Antony's saying few words. Ironically, Brutus's remark to Cassius that "the eye sees not itself/But by reflection...(1.2.52-53) seems all the more true with himself.
Upon hearing of Caesar's death, Antony requests to be allowed to speak to the conspirators so he might learn the reasons for the murder. Antony assures the conspirators that he does not doubt their wisdom and offers to shake hands with them, making it seem as if he is sympathetic toward their motivations for killing Caesar. More importantly, Antony asks for permission to speak at Caesar's funeral, and this request is the root of the disagreement between Brutus and Cassius.
As Brutus is a good-hearted and trusting man, he agrees to let Antony speak, saying, "You shall not in your funeral speech blame us,/ But speak all good you can devise of Caesar,/ And say you do 't by our permission." Thus, Brutus decides to allow Antony to speak at the funeral after Brutus has finished speaking.
Cassius, though, is wary of Antony's intentions and says, in an aside to Brutus, "I know not what may fall. I like it not." Brutus, however, dismisses Cassius's worry.
Similar disagreements between Brutus and Cassius occur at other points in the play. In Act 2, scene 1, the two disagree over whether or not to kill Marc Antony at the same time they murder Caesar. Brutus insists that Antony is "but a limb of Caesar," and will have no power once Caesar is dead. Cassius, on the other hand, does not trust Antony and thinks that he needs to be killed, also. (Similarly, Brutus and Cassius disagree with regard to battle plans in Act 5. Brutus believes that their army should advance to meet Antony's army at Philipi, while Cassius insists that it will be better for their army to rest and let Antony's army tire themselves out by journeying to Cassius's and Brutus's army.)
In the case of each of the above-mentioned disagreements between Brutus and Cassius, Brutus ultimately gets his way. However, though Brutus's plans seem well-thought-out, they prove, once executed, to have been major errors in judgment.