The differences are of two kinds: generic, and character-depicting. Elizabethan drama, technically those dramas written during Elizabeth I’s reign, but more commonly meaning the explosion of theatre life in London in the late 16th and early 17th centuries, certainly drew from the traditions of classical drama, technically Greek and Roman drama, but more commonly referring to the dramas following Aristotle’s definition as laid out in the Poetics, but added genres beyond the strictly “tragic”, such as tragicomedy, romantic comedy, and histories (it should be remembered that Aristotle’s definition of comedy was lost to history). When following “the classic” rules of tragedy, such as “a person in a high place falls”, the Elizabethan playwright interpreted the rules rather loosely(for example, Hamlet is not the highest in rank in that play), and allowed dramatic conflicts that did not necessarily lead to “catharsis.” The characters of Elizabethan dramas were psychological depictions of people of every class; in fact, many scholars attribute Shakespeare (the ultimate Elizabethan playwright) with “inventing” stage characters based on psychological profiles. Finally the make-up, and therefore the target audience, of Elizabethan theatre differed from classical audiences in commercial and sociological ways, and the motives for writing an Elizabethan play were no to win a prize, but to make a living.
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.