I've always been fascinated with katharsis. When I first studied drama, the concept caught my imagination and I think it still does. The idea that a play, or any story, can suddenly come to a combination of understanding and extreme emotion appeals to me.
Aristotle's katharsis is a little different from our catharsis. His dramatic katharsis meant that the play had a suitable ending that fitted with all the particulars of the play: the audience perceived and felt that the tragic hero had met the right and correct final suffering commiserate with his tragic flaw. So katharsis is an internal element to the drama.
In the Elizabeth era katharsis shifted, possibly under the influence of the spiritual focus of the Protestant Reformation. It came to mean that the audience had reached a level of intense identification with the play from which they experienced catharsis of their inner being and had learned through a mimetic experience to eschew certain behavior. [Mimetic: imitating the true value actualized in Heaven and taught on Earth by the inspired poet. Eschew: vigorously stay away from.] So Elizabethan catharsis is an external element to the drama, though internal to the audience.
Hamartia is a term developed by Aristotle. This definition is found at eNotes and is described as...
a character’s flaw or error...hamartia is the tragic flaw of the
protagonist in a given tragedy.
Examples characters that demonstrate this "trait" are Shakespeare's tragic heroes. They are "great" men (meaning that they are accomplished, brave, ethical, etc.) who have a flaw of such enormous proportion that it ultimately brings about their own demise. There are many of these kinds of heroes in Shakespeare's plays, but my favorites are Hamlet and Brutus. Hamlet (in the play by the same name) is called to avenge is father. He is generally regarded as a man of grave indecision. (Now, I have to disagree personally, for I find that he hesitates to kill his father's murderer primarily because to murder a king unjustly, costs the murderer his immortal soul; Hamlet has to make sure the ghost claiming to be his father is telling him the truth.) Hamlet is a man of great character and great passion. Ultimately, he avenges his dad's death, but Claudius (the murderer) is so evil, that he takes out almost everyone else with him.
Brutus from Julius Caesar is also considered a tragic hero (though the play is not named for him). Depending upon what research you read, you may get a different answer (I've seen several), but I believe Brutus' flaw is either his tendancy to trust the wrong people (Cassius, Antony, etc.) or his unrealistic expectations. He wants to save Rome from Caesar's tyranny, but he is surrounded by men who have no such noble intentions. To change the face of Rome as he wanted, Brutus would have needed much more support from other leaders within the empire. He may not have planned well enough or he may have been too impulsive, but either way, I see him as a great man: an man of ethics with a great love for his country. As with Hamlet, he too dies because of his tragic flaw.
Source for eNotes definition: http://en.wikipedia.org/wiki/Hamartia