In Act III, Scene 1, why does Cassius argue against allowing Antony to speak at Caesar's funeral? What reasons does Brutus give for overruling him?

Cassius understands men much better than Brutus. He would have liked to see Antony killed along with Caesar, but Brutus overruled him. In Act III, Scene 1, Antony virtually ignores Cassius and very wisely appeals to Brutus to spare his life and allow him to speak at Caesar's funeral. Cassius is appalled when he hears the gullible Brutus give Antony permission to do so. Cassius appeals to Brutus in an aside as follows:

Brutus, a word with you.
You know not what you do. Do not consent
That Antony speak in his funeral.
Know you how much the people may be moved
By that which he will utter?

Brutus is a philosopher and an idealist. He trusts Antony, whereas Cassius knows Antony is deceitful and not to be trusted. Earlier, he calls Antony "a shrewd contriver." Cassius makes the mistake of persuading Brutus to act as the leader of his conspiracy because only Brutus can make the assassination of Julius Caesar seem like a necessary and patriotic action. Brutus goes further than Cassius anticipated. Brutus tries to apply his high-minded principles to the entire enterprise. He shows himself to be impractical and conceited. He refuses to listen to Cassius or anyone else. He replies to Cassius as follows:

By your pardon.
I will myself into the pulpit first,
And show the reason of our Caesar's death.
What Antony shall speak I will protest
He speaks by leave and by permission;
And that we are contented Caesar shall
Have all true rites and lawful ceremonies,
It shall advantage more than do us wrong.

Brutus is extremely vain about his rhetorical prowess. He is known as an accomplished orator. He is a reasonable man and believes other men can be persuaded by reason. Antony knows this, and he uses it to manipulate Brutus when he tells him,

Friends am I with you all, and love you all
Upon this hope: that you will give me reasons
Why and wherefore Caesar was dangerous.    III.1

Antony doesn't really care about Brutus's reasons, but Brutus swallows the bait. He is a reasonable man and loves to think and talk about his reasons. Antony fully intends to appeal to the crowd's emotions, not their reasons. He doesn't expect much rationality from ignorant men of the lower classes. They are flattered by Brutus, a distinguished man who addresses them as if they are educated, literate, and high-minded like himself. When Antony follows Brutus onto the pulpit, he shows he understands his audience far better. He has an ace-in-the-hole, which Shakespeare's audience does not know about until he reveals it. Antony has Caesar's will concealed under his tunic. It leaves money and lands to all the Roman citizens. When Antony reads this will in Act III, Scene 2, at the end of his funeral address, it serves to inflame the mob to mutiny, which is exactly what Antony wanted.

ANTONY: Moreover, he hath left you all his walks,
His private arbors, and new-planted orchards,
On this side Tiber; he hath left them you,
And to your heirs for ever; common pleasures,
To walk abroad and recreate yourselves.
Here was a Caesar! When comes such another?

FIRST CITIZEN: Never, never. Come, away, away!
We'll burn his body in the holy place
And with the brands fire the traitors' houses.
Take up the body.

Antony provokes the very opposite of what Brutus wanted, which was peace, harmony, understanding, law, and order. Rome becomes a dangerous place for the conspirators. Antony is soon informed by a Servant who just left Octavius:

I heard him say Brutus and Caesar
Are rid like madmen through the gates of Rome.