This is an interesting question and one that is a little difficult to answer because the Renaissance tragic hero--the Shakespearean tragic hero--took on some aspects that were rather different from the aspects defining the Aristotelian tragic hero. By Aristotle's definition, the tragic hero infused the audience with fear and pity while, in the resolution, the catharsis of the drama (Aristotelian dramatic catharsis: the reasonable and natural outworking of the tragic circumstances to a plausible and just resolution) might allow for the hero, like Oedipus, to suffer a fate other than death; you'll recall that Oedipus (like Lear at one stage) was blinded and exiled alone and friendless. In other words, the tragedy need not be so horrible that the hero must die.
On the other hand, by the Renaissance/Shakespearean definition, the tragic hero infused the audience with perhaps more fear and horror than pity (though pity was certainly evoked) while, in the end, the catharsis was seen as a more psychological catharsis in which the audience expended its pent-up impulses toward various forms of pride, greed and revenge by empathizing with the hero who could suffer punishment justly through death. You may recall Macbeth's loveless, friendless and humanly isolated death at the resolution and recognize it as demonstrative of this change in definition. In other words, the tragedy for the Renaissance/Shakespearean hero was so terrible that he must die to satisfy a plausible and empathetic and just resolution.
Though we are digging out similarities, three differences are (1) the objective and type of catharsis and (2) the degrees of fear and pity and (3) the ultimate end of the hero. With this said, how are the two styles of hero alike? They are alike in that both types of hero operate in life out of a basic flaw in their inner character or out of some largely encompassing mistaken idea or action. For Aristotle, the hero is a good, high positioned man of admirable qualities who has a fatal flaw in his character traits or in his understanding: these flaws (character trait or mistaken understanding) lead to ideas, decisions and actions that result in unmanageable tragedy.
For Renaissance dramatists, like Shakespeare, the hero is similarly a good, high positioned man of admirable qualities and with renown who has a fatal flaw in character traits or in understanding. But for Shakespeare, the flaws at work in creating the tragedy may be both a fatal flaw in character traits and (at the same time) a fatal flaw in understanding that together lead to ideas, decisions and actions that result in fatal tragedy. Think again of Macbeth. His subservient love for his wife was one of his fatal flaws in character, while his fatal flaw in understanding led to grievous mistakes that, when combined with his character flaw, resulted in a tragedy so great that he must die as a result.
So while much about these two styles of hero is different, these aspects are the same and each applies to Macbeth:
- high and powerful standing.
- good men but flawed in inner traits.
- flawed in understanding making fatal decisions.
- catharsis of one kind or the other drives the plausible and just resolution of the tragedy.
- the hero is horribly punished for the tragedy he creates.