Antony opens "Julius Caesar" as something of a wild child, during the feast of Lupercal. He is clearly a favorite of Caesar, as he is asked to touch Julia, the wife of Caesar, so that she might conceive. While the conspirators realize he is a man of importance to Caesar, they underestimate him, and he proves to be more than a mere limb of Caesar, who can be rendered powerless with out the figurative head of power. This proves to be fatal mistake in the end. We see Antony come into his own during the funeral speech. As power is divvied up, Antony claims what he feels is his, putting him at odds with Octavian, the young, adopted nephew of Caesar.
As "Antony and Cleopatra" opens, we see an Antony who is in touch with his political power, but the wild child is within sight as he begins this tempestuous affair with the mother of Caesar's son. He continues in this ill-advised affair that clouds his judgement politically and personally. Antony believes his own press so to speak because he is a man who is loved by a goddess, which is how Cleopatra is regarded amongst her people. He, like Caesar, suffers from delusions of grandeur and he, like Caesar, will pay for his hubris with his life.
In "Julius Caesar," Antony is portrayed as the strong general, so full of vigor that a mere touch of his hand can make a barren woman fertile (see Act I, scene 2). He is a brilliant orator and a valiant soldier.
In "Antony and Cleopatra," the brave general has become a lovesick fool. The opening speech by Philo says it all:
Nay, but this dotage of our general's
O'erflows the measure: those his goodly eyes,
That o'er the files and musters of the war
Have glow'd like plated Mars, now bend, now turn,
The office and devotion of their view
Upon a tawny front: his captain's heart,
Which in the scuffles of great fights hath burst
The buckles on his breast, reneges all temper,
And is become the bellows and the fan
To cool a gipsy's lust.
Look where they come:
Take but good note, and you shall see in him
The triple pillar of the world transform'd
Into a strumpet's fool: behold and see.
When Philo calls Antony "the triple pillar of the world," he is probably referring to the triumvirate formed after Caesar's assassination. Antony was once one of the three most powerful men in the Roman world. Now he has become a "strumpet's fool."
I'm not familiar with "Antony and Cleopatra", but I can help you with how Antony is protrayed in "The Tragedy of Julius Caesar". Antony is see as incredibly loyal to Caesar; in fact, Cassius suggests killing Antony because Antony is so loyal to Caesar that he might seek revenge. Antony is referred to in the play (and was historically) a partier and a playboy. During Act III, Antony uses his intelligence and power of speech (rhetoric) to sway the commoners to riot and drive Brutus and Cassius out of town, seemingly to avenge Caesar's death. At the beginning of Act IV, Antony has changed from a man who is looking out for the plebians and his friend to a war general, looking to cut the plebians off from Caesar's inheritance. Antony is either very changeable or two-faced--from the historical notes, we can assume he is two-faced and more interested in personal gain than the good of the Republic.