Brutus wants to spare Antony because he does not want to shed any more blood than is strictly necessary. According to his own viewpoint, the only necessary death is Caesar's. Actually, he does not want to kill Caesar either; he hates the idea of murder. At the same time, though, he is persuaded that Caesar must die for the greater good of Rome, as there is danger that he might end up amassing too much and lifelong power.
Cassius is much more of a political realist than Brutus and realizes that Antony, as Caesar's close follower, does pose a threat to the conspirators. However, Brutus shrugs off the idea. His naive idealism leads him to believe that somehow everything will come out all right once Caesar is dead. When Antony in an excess of grief first comes to the conspirators after Caesar's murder and demands that they kill him as well, he replies simply, "O Antony, beg not your death of us," and seems to expect Antony and the rest of the Romans to understand why Caesar was killed and that it was in fact an honorable thing to do. Of course Brutus pays dearly for underestimating Antony when he not only spares his life but even allows him to make a full speech at Caesar's funeral. Antony, a master manipulator, seizes the chance to turn the Roman people entirely against the conspirators.