There are, of course, many differences between modern and Greek tragedy. Here are several of them:
1. Greek tragedies involve a Chorus. The Chorus often provides background detail, poetic elements, and summaries of events in between scenes. Sometimes the Chorus serves as a type of moral judge over characters' actions and words; Sophocles' Antigone is a good example of this type of Chorus. Modern tragedies do not have a chorus or really anything similar to it.
2. Modern tragedies in written form provide readers with stage directions and usually detailed descriptions of the play's sets. While the Greeks did use masks to show character change, they performed their tragedies on a bare stage, where elaborate sets would have been insignificant even if they were available.
3. Most importantly, while Greek tragedies feature their heroes or heroines realizing their flaws and sometimes repenting of them, modern tragedies often do not. In Death of a Salesman, a modern play, Willie Loman commits suicide, but it is not the noble death of a Greek tragic hero; rather, he takes his life out of hopelessness. Admittedly, many Greek tragic heroes commit suicide, but they do so after realizing that their flaw has brought down them and the ones they love--their deaths would have been viewed as moral and noble by an Ancient Greek audience.
Probably the biggest difference between a modern tragedy and a classic Greek tragedy is the tragic hero. In a Greek tragedy, the hero is usually someone of elevated status. Take Hamlet or Macbeth for examples. Both men are from the nobility. Another key to the classic tragic hero is that they have some kind of "tragic flaw" that ultimately leads to their downfall. Let's look at Macbeth again. He has unrestrained ambition, and that leads him to commit all kinds of terrible acts in pursuit of more power. For Macbeth, the end justifies the means.
Comparatively, a modern tragedy's hero character is much more of a common man. Additionally, the tragic flaw that causes the character's downfall is either not present or isn't as central. The character's downfall is usually the result of societal pressures/events, and the character often dies completely unrecognized as a hero. The previous answer used Willie Loman as an example. He's a solid choice. I'll pick one from film because more people will see a movie than a play. I like Jack from Titanic as a tragic hero. He's a common man. He doesn't come from the wealthy ruling class. His death is just about completely out of his hands. His actions are guided by the changing situation. He's not in control of anything. Finally, he dies unrecognized by anyone (other than Rose) for his heroic acts.