This is a great question and one the can be answered in many different ways. With that stated, I think one of the major difference between ancient and modern tragedies is that in modern tragedies, the main character can be an ordinary man. In ancient tragedy or Greek/Roman tragedy, the main character had to be noble.
If you think about the great characters of Greek tragedy for instance, they were all from the nobility: Oedipus, Antigone, Agamemnon, and many others. When it comes to modern tragedies, we have people like Willy Loman in the Death of a Salesman. Even his name suggests something common.
Another difference is that in ancient tragedies the gods or divine matters are much more pronounced. For example, in the story of Oedipus, we read of prophecies, seers, and the intimate work of the gods in the lives of people. In modern tragedies, there is little of the divine; people take matters into their own hands.