How ancient Greek Drama is different from our contemporary experience at the movies or at the theater.

The biggest difference between Greek drama and modern day entertainments is that the former was largely religious in nature. Dramas weren't staged in Ancient Greece purely, or even mainly, for the purposes of entertainment. They took place as part of elaborate religious rituals in which the whole community was invited to participate. Attending the modern-day theater or going to the cinema to catch a movie doesn't have quite the same communal value. It's certainly not a religious experience, nor is it intended to be.
The scholarly consensus is that Greek tragedy emerged out of rituals in which Dionysus, the god of fertility—and later theater—was worshipped by devotees wearing masks. This is widely held to form the basis of the mask-wearing tradition in both Greek tragedy and comedy.
In terms of subject matter, it is no accident that Greek drama often explored the relations between gods and mortals or that it also provided an in-depth commentary on religious practices. All surviving Greek plays draw extensively upon a rich mythological heritage in which the immortals play such a crucial part. The gods aren't there to entertain but to remind the audience of their sacred duty towards the immortals and to the importance of paying them their due homage.

Ancient Greek drama was a central part of the social and entertainment lives of the Ancient Greek people. Each year, there was a drama competition held as part of the Festival of Dionysus. The plays would each be performed for the large audience and one playwright would win recognition for having written the best play. Winning this prize was a great honor, even though it was not accompanied by a large financial sum or other physical object: the prize was more about the reputation of the playwright and bragging rights. The community would gather to watch the plays, and in the case of tragedies, would experience catharsis. This means that through watching the tragic fall of the hero, the audience would purge their emotions, feeling cleansed by the experience. The community also saw itself represented in the Chorus, which provided both entertainment and social commentary during the plays. Finally, as part of the Festival of Dionysus, one of the Greek gods, the drama festival and competition were inherently religious rites. The masks worn by characters in the play were also part of a religious ritual, in addition to serving the practical purpose of allowing actors to play more than one role.
In some ways, an experience of going to a contemporary film or play is similar because our dramas are so influenced by the Ancient Greek classics. We do gather communally in theaters and sometimes experience catharsis when watching a movie or play. We do still recognize "the best" of the products in awards shows like the Oscars and the Tonys. We do still respect writers and directors who are involved in the creation of "the best" movies and plays. The major difference is probably that our movie-and-theater-going experiences are really not linked to religious practices, in most cases. Also, we do not gather together as one community to watch one or a few plays on a given day. We have more options for watching entertainment, and we are more widely dispersed as we enjoy these productions.