While both the Greek and the Elizabethan tragic hero have tragic flaws and make errors in judgment that enable or lead to their downfalls, the Elizabethan tragic hero is generally to blame for his ruin, whereas the Greek tragic hero is often a victim of fate. Oedipus, in Oedipus Rex, for example, is a tragic hero who suffers from the tragic flaw of pride. However, his fate was decided even before he was born; the oracle at Delphi, Apollo's mouthpiece, told Oedipus's father that he would have a son that would kill him and marry his wife. Oedipus's pride, which leads him to believe that he can avoid the prophecy, is what actually enables it to come true, but our sense that one cannot defy the gods' will is a significant part of his story. Shakespeare's Macbeth, on the other hand, seems much more responsible for his own demise; it is the result of his choices, not the result of fate. He chooses to murder the king in order to usurp the king's role—he was not fated to do so—and this action leads to his ruin. By the Elizabethan era, people preferred to believe that they had control over their lives rather than believe that something arbitrary—like fate—was directing their actions.

What are some differences between the heroes in Elizabethan tragedies and Greek tragedies?

In Greek tragedy, a hero is carefully defined – falling from a high place, tragic flaw, catharsis, etc. – as defined and illustrated by Aristotle in The Poetics.  By Elizabethan times the term “tragedy” had become somewhat looser, describing a genre in which persons suffered various inequities (some of which – Macbeth and King Lear, for example – still fulfilled the Greek definition, and some which do not – Romeo and Juliet and Troilus and Cressida, for example).  These tragic heroes were more dramatic constructions to tell a story on the stage, rather than a quasi-religious icon of a philosophical world view.  Elizabethan heroes were more complex in their motives, in their psychological make-up, in their social standing, part of the "commodity" of theatre, rather than part of the ritual to honor Dionysus, as Greek tragedies were. Finally, the very term "hero" becomes rather ambiguous in Elizabethan drama, because whole societies were not necessarily affected, as kingdoms were in Greek tragedy.