Why does Cassius want Antony killed, and why does Brutus wish to spare his life?

Cassius is a better judge of human character than Brutus, as the events in the play reveal. Brutus is an idealist. He has an honest, generous character and expects other men to be the same. When he takes over the titulary leadership, he wants the conspiracy to appear to the public to be a patriotic and necessary deed. Cassius is a selfish, greedy man who expects other people to be like himself. When he suggests Mark Antony should be killed along with Julius Caesar, he foresees, correctly, that Antony could be extremely dangerous. Brutus and Cassius both state their opinions of Antony in Act II, Scene 1.

CASSIUS: I think it is not meet
Mark Antony, so well beloved of Caesar,
Should outlive Caesar. We shall find of him
A shrewd contriver; and you know his means,
If he improve them, may well stretch so far
As to annoy us all, which to prevent,
Let Antony and Caesar fall together.

Antony does in fact prove to be a shrewd contriver. He knows how to manipulate Brutus, and then shows he knows how to manipulate a whole crowd of lower-class Roman citizens who all listen to Brutus's funeral speech, support Brutus, and favor the assassination of Julius Caesar.

When Cassius suggests killing Antony in Act II, Scene 1, Brutus dismisses the proposal. Brutus is a good and wise man in many respects, but he has his faults. He is a bookish, solitary man wrapped up in his own ideas. He assumes he knows more than any other man in Rome. He has a reputation for integrity and patriotism, largely because of his family background, and this is of great importance to him. He wants the masses to think of him as a hero after the assassination. In reply to Cassius's proposal to kill Antony, he says,

Our course will seem too bloody, Caius Cassius,
To cut the head off and then hack the limbs
Like wrath in death and envy afterwards;
For Antony is but a limb of Caesar.
Let us be sacrificers, but not butchers, Caius.
We all stand up against the spirit of Caesar,
And in the spirit of men there is no blood.
O, that we then could come by Caesar's spirit,
And not dismember Caesar! But, alas,
Caesar must bleed for it! And, gentle friends,
Let's kill him boldly, but not wrathfully;
Let's carve him as a dish fit for the gods,
Not hew him as a carcass fit for hounds;
And let our hearts, as subtle masters do,
Stir up their servants to an act of rage
And after seem to chide 'em. This shall make
Our purpose necessary and not envious,
Which so appearing to the common eyes,
We shall be call'd purgers, not murderers.
And for Mark Antony, think not of him,
For he can do no more than Caesar's arm
When Caesar's head is off.

When Brutus speaks to the mob in Act III, Scene 2, he tries to make them think he and all the conspirators were public-spirited "sacrificers." When Antony displays Caesar's shredded, bloody cloak and mangled body, they serve as evidence that most of the conspirators were "butchers, not sacrificers" and that they "hewed Caesar as a carcass fit for hounds." In truth, most of the thirty or more assassins were motivated by hatred, rage, fear, and panic, not the noble motives Brutus attributes to them.