In the early days of drama, the Church was primarily involved. This would change, sometime during the 16th Century. The English Renaissance arrived in England close to one hundred years after Italy's Renaissance. This "rebirth" opened up new possibilities to those who worked with the arts.
The Renaissance brought with it the great drama of classical Greece and Rome.
These great dramatists would include the works of Aristotle, Sophocles, and Seneca among others. Drama changed enormously.
The first plays to follow the mystery, miracle and morality (religious) plays included a conflict centered around three principle characters: the hero, heroine and villain, which is not seen in modern drama. Shakespearean drama followed the beginnings of early dramatic presentations. With the writings of authors like Christopher Marlowe and Shakespeare, historical plays became very popular because the audience was enthralled with England's "heroic past."
Shakespeare was able to manipulate his audience into feeling "terror and pity" in his tragedies (e.g. Hamlet, Romeo and Juliet, Macbeth, etc.). Shakespeare's comedies were also popular, such as A Midsummer Night's Dream. Ben Jonson was another writer who enjoyed "poking fun at society." Drama took on a new "face" with the introduction of materials that made the audience laugh.
And although today we may not be able to appreciate or understand risque elements of the dialogue of that time because of the changes in language since that era, plays at that time were full of sexual innuendo: "off-color" jokes that were hidden between the lines. These kinds of plays were described as "bawdy."
However, in modern drama, the story's villain is "internalized," something within a major character; in some cases, he/she became his/her worst enemy. This occurred around the turn of the twentieth century.
Modern drama no longer had its heroes, heroines and villains of days gone by—the subjects of the stories were now ordinary people one might meet any day on the street. Henrik Ibsen's modern drama A Doll's House is an example of a modern drama.
With modern drama:
...the political -- even radical -- implications of the work of playwrights...
were evident. Authors of modern dramas included Ibsen, George Bernard Shaw, William Butler Yeats, and Leo Tolstoy among others. Realism was a major distinction between the drama of Shakespeare's time and modern drama. There were elements of politics included in drama, as well as opportunities for the playwright to address concerns at the forefront of the public's attention. Drama allowed its player to "air" their concerns in a public forum. In additiona, the theater became more modern in its layout, as did popular methods of direction.
From a global perspective, then, the very term modern drama must reflect the specific cultural and political history that has shaped it, locally.
Linquistically for one, slang is more prevalent vs. the thoughtfulness of Shakespearean verse, then there is the antiquated English dialect. Social status is a possibility, married parents vs divorced... Mixed races? Would Shakespear have ever written something like that? Look for taboos of old to find your differences.