In Act 4 of Julius Caesar, how do the relationships of Antony-Octavius and Cassius-Brutus compare?

Antony and Octavius vs Brutus and Cassius In Act IV, how does the relationship between Antony and Octavius compare to and contrast with that of Cassius and Brutus?

The two relationships are alike in that they are both based on mutual need. Antony and Octavius need to work together to defeat Cassius and Brutus; Cassius and Brutus have no chance to prevail against Antony and Cassius if they don't settle their differences and stay united in their war against their enemies.

The relationnships are alike also in that each one exhibits a leader and a follower. Antony's word takes precedence over that of Octavius. Cassius bows to Brutus's will even when Cassius thinks Brutus is wrong.

How are these relationships different? The best response here, I think, has already been mentioned. Brutus and Cassius do have a personal relationship whereas Antony and Octavius do not. Brutus and Cassius have been through a horrendous experience together. Together they have plotted and and acted to murder Caesar. Together they have been driven from Rome, running for their lives.

Because of these facts, there is another difference that is understandable, considering human nature. The relationship between Cassius and Brutus is much more emotional. They argue violently, and the argument gets personal.

The relationship between Antony and Octavius is cool and professional. For example, when Antony and Octavius have a difference of opinion as to cutting Lepidus out in dividing the spoils of war, Octavius lets Antony have the last word. Octavius raises some objections, but does not argue with him.

Antony and Octavius are more formal with one another--not truly friends.  They are put together in the name of vengeance--Antony's friend and Octavius' relative Julius Caesar had been murdered.  In fact, the connection between the three--Antony, Octavius, and Lepidus is strained.  The first two do not recognize Lepidus as anything more than one to run errands and do their bidding, but they also seem to question where they stand with one another.  Antony was Julius' right-hand man, but Octavius is a blood relative and seemingly the right choice for the "throne" had this been a dynasty and not a republic.  There are issues of mistrust and tension between these two.

On the other hand, Brutus and Cassius are truly friends.  They respect one another and treat each other as brothers.  Aside from the spat they have over accepting bribes and engaging in dishonorable behavior, they are bonded in their act for the betterment of Rome.  Brutus is an honorable man who believes all the way to his deathbed that he did what he did for Rome, not for personal advancement.  Cassius, perhaps, was in it more for personal reasons...he admits through his actions and thoughts in the play that he was jealous of Julius Caesar.  Whatever the reasons for each man participating in the murder of Caesar, it does not take away from the depth of their friendship.

In Julius Caesar Act 5, Scene 1, how do Brutus and Cassius' argument differ from Octavius and Antony's?

I'm assuming you mean scene 1 of Act 5, where Octavius and Antony are arguing about battle strategies, then later in the scene, Brutus and Cassius are discussing what will happen if they are defeated?

If that is the case, then there are definitely differences as I mentioned above.  Octavius and Antony are butting heads over not just who should take which flank of the battle, but more importantly, who is truly in charge of this triumvirate and their army.  Octavius, as Julius Caesar's heir, does not think he should take orders from Antony, and so we can see that this idea of a 3-man rule of Rome (Lepidus isn't even around anymore) just isn't going to work.  Octavius refuses to listen to Antony's strategies, and so there is some heated "discussion" happening here.

Brutus and Cassius, on the other hand, aren't really arguing, but rather determining what they will do if they are defeated by Octavius and Antony. Brutus had formerly condemned his father-in-law's suicide (Cato), who killed himself to avoid having to submit to Julius Caesar. Cassius is questioning Brutus as to whether he is willing to submit to Octavius' authority in the same situation, and Brutus replies that no, he understands now why Cato did what he did, and he plans to follow the same path if the battle goes against them.

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Compare and contrast the characters Octavius and Antony in Julius Caesar.

Antony is always characterized as a man who likes physical activities and physical pleasures. He is first shown in Julius Caesar preparing to participate in a footrace. He likes partying and having fun. This characteristic remains consistent throughout Shakespeare's Julius Caesar and his Antony and Cleopatra. It is Antony's hedonism that leads to his downfall at the hands of Octavius when they have their final falling out in Antony and Cleopatra.

Octavius is a much younger man than Antony. Without Antony, Octavius would never have been able to succeed Julius Caesar as his heir. It was Antony's famous funeral speech that turned things around and enabled Octavius to head the triumvirate which eventually overthrew Cassius and Brutus at the battle of Philippi, recording in the last act of Julius Caesar. Without Antony, Octavius would not have been victorious in that battle because he lacked military experience. He appreciates the help and guidance he received from the older man and is willing to share the rule of the Roman Empire with him, until Antony's lasciviousness and negligence leads to a rift which results in the death of Antony and Cleopatra, as dramatized in that play by Shakespeare.

Octavius' character is just about the direct opposite of Antony's. Octavius is very serious, intelligent, calculating, cerebral, abstemious, and practical. This is why he wins sole rulership of the Roman Empire in the end. In Antony and Cleopatra, there is a scene in which the two men are compared by a soothsayer, who very accurately evaluates Octavius and Antony and their relationship.

ANTONY.
Say to me,
Whose fortunes shall rise higher, Caesar's or mine?

SOOTHSAYER.
Caesar's.
Therefore, O Antony, stay not by his side:
Thy demon, that thy spirit which keeps thee, is
Noble, courageous, high, unmatchable,
Where Caesar's is not; but near him thy angel
Becomes a fear, as being o'erpower'd: therefore
Make space enough between you.

ANTONY.
Speak this no more.

SOOTHSAYER.
To none but thee; no more but when to thee.
If thou dost play with him at any game,
Thou art sure to lose; and of that natural luck
He beats thee 'gainst the odds: thy lustre thickens
When he shines by: I say again, thy spirit
Is all afraid to govern thee near him;
But, he away, 'tis noble.

ANTONY.
Get thee gone:
Say to Ventidius I would speak with him:--

[Exit SOOTHSAYER.]

He shall to Parthia.--Be it art or hap,
He hath spoken true: the very dice obey him;--
And in our sports my better cunning faints
Under his chance: if we draw lots, he speeds;
His cocks do win the battle still of mine,
When it is all to nought; and his quails ever
Beat mine, inhoop'd, at odds. I will to Egypt:
And though I make this marriage for my peace,
I' the East my pleasure lies.        (Act 2, Scene 3)

There is no better comparison between Octavius and Antony to be found anywhere than in this exchange between Antony and the Soothsayer. Many readers will relate to Antony's thoughts and feelings. There are some people who make us feel inferior, even when they are supposedly our friends. The Soothsayer gives us good advice when he says, "Make space enough between you."

Macbeth refers to Banquo as a person like Octavius in Shakespeare's play.

There is none but he
Whose being I do fear; and under him
My genius is rebuked, as it is said
Mark Antony's was by Caesar. .    (Act 3, Scene 1)

By "Caesar," Macbeth means Octavius Augustus Caesar, who became the Roman Emperor and sole ruler of the empire after defeating Antony in a battle at sea and in a land battle in Egypt. 

Mark Antony tries to stay away from Octavius, but perhaps he needs someone like Octavius to exercise some restraint on his reckless and licentious character. In Egypt, Cleopatra is too much like Antony. She encourages him to be lazy and self-indulgent. His close associates deplore the way Antony spends all his time fornicating and partying and neglects the responsibilities of running his half of the Roman Empire. Octavius is remarkably tolerant for some reason, perhaps because he is grateful for the help he received from Antony immediately after Julius Caesar's assassination. In the end, it seems inevitable that the older, dissolute Antony, captivated by the cunning, voluptuous Cleopatra, would be defeated by the younger, more disciplined, more selfish and opportunistic Octavius. It is another case of youth winning out over age.

What are the main differences between Antony and Octavius in Julius Caesar?

Beyond the issues of age, one of the major areas of contrast would be the manner in which they make decisions.  Antony, the older and wiser soldier and politician, makes decisions based upon his political/military experience, showing his loyalty to Caesar as well as his strategy to centralize power by supporting Octavius and marginalizing Lepidus in Act 4.  In this way, Antony demonstrates his knowledge and distrust of the motives of others as well as his willingness to be "cut-throat" in forwarding his own agenda.  Octavius appears very much as the young, privileged aristocrat whose decisions show his youthful idealism, inexperience, and stubborn will.  He is not as dismissive of Lepidus as Antony is, but agrees to follow his lead until
Antony tries to assert his authority in the heat of battle where Octavius rejects his instructions in order to show Antony his independence and rank as Caesar's obvious heir.  In this way, Octavius could be seen as straight-forward and impetuous while Antony is strategic and calculating.