Actually, Antony doesn't deliver a public funeral oration for Brutus as he had for Caesar. His short speech about Brutus is delivered on the battlefield, and heard by few present, after Antony and his forces have prevailed at the play's conclusion.
Why Antony spoke so approvingly of Brutus most likely relates to Shakespeare's structure in developing a tragedy. The tragic hero is first established as a good man who enjoys the respect of others, then he falls, destroyed by a fatal flaw within him. He struggles against his own destruction, but he does not prevail, and he dies. At the conclusion of the play, in order to emphasize the tragedy of the hero's fall and destruction, Shakespeare then reminds the audience of how good and great his hero once had been. This seems to be the reason for Antony's assessment of Brutus as having been "the noblest Roman of them all." But this isn't enough for Shakespeare's purposes. Antony continues to elevate Brutus in his superiority to others:
His life was gentle, and the elements
So mixed in him that Nature might stand up
And say to all the world, "This was a man."
The reference to the "mixing" of "elements" refers to some Elizabethan beliefs in regard to human physiology, but the message is clear: Brutus was exceptional. Therefore, his destruction is especially tragic. Antony's closing speech about Brutus is a literary convention in Shakespearean tragedy.
Antony realizes that Brutus, unlike the other conspirators, participated in the murder of Caesar for what Brutus felt was for the good of Rome. Cassius and the others were motivated by self-interest. So he praises Brutus. Here's something else to consider. Antony demonstrated that he can rouse a crowd to action with his words--just as he did immediately after Ceasar's murder. The public listens to Antony. The public also respected Brutus (until Antony convinced them that he was NOT an honorable man). Perhaps Antony is behaving like a typical politician by praising a rival who is no longer a threat.
He still respects Brutus for his courage and his honor and think that whatever that he did is for the good of Rome and not for his own interest so he praise him. Antony calls Brutus the "noblest Roman of them all," but his words thereafter suggest that the "all" in question includes only the conspirators against Caesar.