Numerous comparisons and contrasts exist between classical and modern drama. I'll deal with a small part of your question.
One of the most significant contrasts between classical drama and modern is the difference in the protagonists. Classical tragedy, for instance, involves royalty, the elite. The idea was that for a character to have a great and far-reaching influence over society he/she had to be in a position of great power and authority.
In contrast, modern drama often uses common people as protagonists. John Synge, for instance, uses Irish peasants in his tragedy, Riders to the Sea, even though the play imitates Greek tragedy. In modern drama, tragedy can be indicative of a society, without totally involving and disrupting that society.
The first issue here is that classical drama was not uniform. Greek tragedy and Old Comedy were radically different from most modern dramas, while Greek New Comedy, Roman Comedy, and some Roman tragedies were closer in style to modern drama. Of course, modern drama is also tremendously varied, with some modern playwrights borrowing elements such as the chorus or integration of singing and dancing into their staging.
Greek Tragedy and Old Comedy differ from much of modern drama by being written in verse. Both genres have a limit of three actors, who play multiple roles. Actors wear masks and stylized costumes (as they also do in Japanese Noh drama), and female roles are played by men as they are in Shakespeare and most Elizabethan drama. The chorus sings and dances (as do choruses in opera and musical theater). Classical drama was presented in amphitheaters, with a circular orchestra and the audience seated in rows built into a hillside surrounding the performance area, while recent (19th through 21st century) drama often uses proscenium stages. Ancient drama was performed outdoors using natural light while contemporary drama is performed indoors using artificial light.
Contra another answerer, while ancient tragedy had noble protagonists, Old and New Comedy and Roman Comedy had protagonists from all classes, including poor farmers, shopkeepers, soldiers, and slaves as well as wealthier protagonists. The "mixed" genre combining rustic comedy with elevated tragedy, though, was mainly an early modern invention, although some of Euripides' plays anticipate this.