Whereas the plots of Greek tragedies and Shakespearean tragedies can be fairly similar, consideration of the actors and staging will yield some of the main differences.
Greek tragic actors wore masks that covered their entire faces, whereas Shakespeare's players did not. Greek tragedies also had a smaller number of actors who spoke in a single scene than in Shakespeare’s plays. In a typical scene from a Greek tragedy, it is fairly rare for more than two actors to speak to one another. Shakespeare’s tragedies also lack the twelve to fifteen member chorus found in the earlier Greek tragedies.
The theaters themselves were also different. How much of a stage was present in ancient Greece is a matter of debate. The primary acting space in a Greek theater seems to have been the orchestra, a large circular space that frequently had an altar in the middle of it. This is not the case in Shakespearean theater. Also, Greek plays were always staged outdoors and during the day. Again, Shakespearean tragedies could be performed in indoor theaters.
We should also note that Greek tragedies were performed as part of religious festivals devoted to the god Dionysus. Shakespearean tragedies do not have this religious alignment.
Further, for the ancient Greeks, honoring the gods via the plays' plots was extremely important. For example, one major theme of the popular Sophocles drama, Oedipus Rex, is that one simply cannot outsmart the gods. For a human being to think that he can escape the fate the gods have laid out for him or that he can avoid a prophecy delivered by one of Apollo's oracles is the ultimate in hubris. Such overweening pride almost always results in terrible punishments for mortals. For Shakespeare's audience, however, such a focus would be much less palatable. His audience wanted to feel like they had some agency in their lives, that they made legitimate choices (not simply taking actions that were preordained by the gods somehow) that actually made a difference in their futures. The idea that they were not in control of their own destinies was not a popular one. Therefore, Shakespeare's plays tend to rely more on a hero's free will rather than his fate (although in some plays, like Macbeth, there is a tension between these ideas).