This is an interesting question, because though the playwrights of both time periods certainly considered their religious views and those of thier audiences to be common source material for creating their plays, the actual experiences of attending the theatre were probably very different.
To begin with, both types of theatre (Greek and Elizabethan) came out of events created to support each era's religious practices. In Ancient Greece the first theatrical performances were in honor of Dionysus and in medieval England (just before the Elizabethan era), plays were presented by the Catholic church as a way to teach an illiterate congregation the lessons of good Christian behavior.
Both theatrical legacies grew from their early days. Greek theatre developed into the first examples of Western Drama, setting the standards we still refer to for Comedy and Tragedy. Yet, the performances of plays maintained some of their connection to religion. Every year in Athens a contest was held, still in honor of Dionysus, at it was at these event that the famous playwrights of Ancient Greece competed and offered their newest works.
In Elizabethan England, on the other hand, the theatre that grew out of the Miracle and Mystery plays of medieval days moved off the streets and into playhouses. And, with this move, they became wholly secular events. A day at the theater could be a bawdy affair, and, though women could attend performances, they could not perform as actors. This was considered a lewd profession. Bear baiting, gambling and other less than pious activities often accompanied Elizabethan theatrical performances. The stricter sects of Christianity (Puritans) protested the unseemly world of the theatre and had theaters closed for a time in 1642.
So, though Ancient Greek and Elizabethan theatres both began as outgrowths of the religious observations of their day, the Greek theatre maintained its connection, while Elizabethan theatre became a wholly secular affair.