What is the difference between Shakespearean tragedy and Greek tragedy?

There are many important differences between Greek tragedy and Shakespearean tragedy. 

Greek tragedy was performed as part of a religious festival (like a church christmas play) - so the stories were already known to the audience, and everyone knew what was going to happen next. 

Elizabethan theatre was commercial entertainment (people paid for their seats - like in a cinema). The stories were usually new, and an element of suspense was nearly always present. 

Greek actors wore elaborate costumes and masks, and parts of the dialogue was sung (parts were even danced). Murders, fights and battles had to take place off-stage (a character would tell the audience what was happening) - as usually happens in opera or a ballet. 

Elizabethan actors wore ordinary clothes (though they might be 'in period' for a historical play). They could scuffle, fight - even 'die' - onstage. (Shakespeare has Tybalt die onstage in Romeo and Juliet, to good effect). 

Because Greek drama was semi-offical, Greek playwrights tended to be highly respected public servants. Most Greek plays take a broadly politically conservative stance (though the best plays can be quite subtle in the points they make). 

Elizabethan players were seen as anti-establishment (they were called 'masterless men'). Many Elizabethan plays are critical of official government positions - though there was rigorous state censorship to make sure they never went too far. Shakespeare's plays, however, consistently adopt a position supportive of the government (and he was far from a "masterless man", being one of the King's Men). 

Another difference is that in Greek tragedies logic over rules emotions (characters try to find out the truth and how things really happened) and in Shakespearean tragedies emotion over rules logic (characters worry about their emotions and what they think is going on rather than finding out the truth).

In Greek tragedy, the chorus is always present on stage as a commentator; in Shakespeare choruses only introduce the scene or the play (see Henry V, Romeo and Juliet, Henry IV Part II, and Pericles for examples)

Greek tragedies usually have one continuous simple plot; Shakespeare's plays have complex plots often involving intertwining subplots (e.g. King Lear)

Common Features of the Shakespeare Tragedies

The Shakespeare tragedies share a number of common features, as outlined below:

• The fatal flaw. Shakespeare’s tragic heroes are all fundamentally flawed. It is this weakness that ultimately leads to their downfall. 
• The bigger they are, the harder they fall. The Shakespeare tragedies often focus on the fall of a nobleman. By presenting the audience with a man with excessive wealth or power, his eventual downfall fall is all the more tragic. 
• External pressures. Shakespeare’s tragic heroes often fall victim to external pressures. Fate, evil spirits and manipulative characters all play a hand in the hero’s downfall.

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