One important way that modern drama differs from classical (or Shakespearean drama) is the notion of the tragedy of the common man. In Aristotle's rules of tragedy, the classical hero must be of noble birth. He (or in some cases, like Antigone, she) doesn't need to be a king, but he should be of elevated social status. Because of this, his fall will be all the more devastating when it comes. Notice that Shakespeare's tragic characters are all nobility.
Yet Arthur Miller changed this forever with his essay, "Tragedy and the Common Man". He postulated that the life of an average person can be just as tragic as that of a noble, and that an audience would identify with the character, perhaps even more so. The greatest example of this at work is probably Death of a Salesman. In that play, Willy Loman is as poignant a character as any classical tragic hero, because in him, most people see an echo of themselves.