Greek drama evolved in the context of religious festival. This alone serves as a critical factor to be aware of: modern drama doesn't have that religious dynamic that was often built into the Greek context. You see this especially in the case of Greek tragedy , which often depicts the...
famous stories of Greek mythology. Structurally as well, Greek and modern drama are distinct from one another. Indeed, as historian D. Brendan Nagle points out,
In the earliest plays there was only one actor and the chorus. Aeschylus increased the number of actors to two and Sophocles to three, and thereafter this remained the standard number. (The Ancient World (5th ed), Upper Saddle River, NJ: Prentice Hall, 199)
Thus, Greek drama tends to rely on its use of a chorus. Furthermore, the limited number of actors required the same actor to play multiple roles (while further limiting how many characters can appear on stage at any one time). Meanwhile, costuming and the use of masks were significant parts of Greek drama:
The masks were a necessity because the size of the theater made it difficult to see the faces of the actors, and because they also permitted the actors to switch roles quickly. (Nagle, 199)
This reliance on masks as a key element of costuming has no equivalent in modern theater.
Finally, Greek drama had a competitive context that modern drama lacks. Audiences would not attend the theater to watch a single production, as they usually do today: rather they would watch a series of productions, which would be judged against each another. While modern drama does have awards and honors which various productions might be in competition for, these awards are not as embedded directly within the theater-going experience as they were in the Greek context. They are entirely supplemental to the dramatic experience. (The same cannot be said in Ancient Greece.)