In addition to the differences outlined in the answer above, one of the major purposes of classical drama was to glorify the gods, and this purpose is missing in much of Elizabethan drama. For example, in Oedipus Rex, Oedipus learns that it is not possible to escape fate—specifically the prophecies originating with Apollo and delivered via his oracle that condemn Oedipus to, one day, kill his father and marry his mother. In fact, it is by attempting to defy fate that Oedipus fulfills the prophecy that he will kill his father and marry his mother.
However, by the reign of Elizabeth I, people no longer wanted to think of themselves as the pawns of gods or of fate; they wanted to believe that their decisions had meaning and significance, and that they could direct the course of their own lives. Therefore, Elizabethans preferred their tragic heroes' downfalls to be their own responsibility, resulting from personal flaws, rather than from some external and uncontrollable force(s) acting on them. As a result, we have characters like Macbeth and Hamlet, who end in ruin because of their own shortcomings, and not because fate or the gods demanded it.