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.