The Greeks built a variety of different theaters, by no means all of which were the type of semi-circular, open-air arrangements we now associate with classical theater. The Odeum of Pericles, for instance, built in Athens in 440 BCE and believed to be the first indoor theater, was a square building which seated an audience of around 4,000, spread around all four walls with a stage in the middle.

The theater of Dionysus provides the pattern for outdoor Greek theaters, a monumental stone semi-circle carved into a hillside, with a flat orchestra (dancing or performance area) and behind it a scene-building, originally wooden, later replaced with a stone building that had projecting wings.

The Romans began to build theaters in the middle of the third century BCE, basing them on Greek models. The most obvious difference was that the began by building every part of the theater from wood. These temporary wooden structures could be fantastically elaborate, with several tiers of seating for 70,000-80,000 people. In the middle of the first century BCE, the Romans began to build theaters in stone. The stone Theater of Pompey was built in Rome in 55 BCE. It was similar to a Greek theater, except for the corridors underneath the seating and the shaded walkways on the top level, both common features of Roman theaters.

Perhaps the principal distinguishing characteristic of later Roman theaters built in stone or brick, was the much higher scene-building, generally several stories tall and built to about the same height as the top level of seating. These made it possible to use ropes secured to the top of the building to cover part of the seating area with fabric, creating a temporary partial roof for the theater and shading audience members from the sun.

Because Greek drama influenced Roman drama, there are many similarities between the two. Both Greek and Roman drama were divided into tragedies and comedies, with tragedies usually set in a distant past, being highly stylized, and addressing great and heroic themes. 

Greek drama was originally divided into three genres, tragedy, Old Comedy, and satyr plays. All of these were performed in outdoor amphitheater-like edifices with seats carved into semi-circular hillsides sloping down to a circular orchestra (literally "dancing place") where the choruses sang and danced, behind which was an acting space and a structure with painted scenery. 

A major fourth-century Greek innovation was New Comedy, exemplified by Menander, which was realistic, concerned with every day contemporary life, and lacking a chorus. Roman comedy, such as the famous plays of Terence and Plautus, consisted mainly of adaptations of Greek New Comedy, just as the tragedies of Seneca were based on Greek models. 

Roman theaters were quite similar to Greek ones, though they could be freestanding buildings rather than built into hillsides. The orchestra was normally semi-circular rather than circular and there were more elaborate stages. 

I am not sure if you are talking about the structure of the building or the plays themselves. I think both have had an impact on the modern theater, in terms of the building itself. However, Greek plays are still performed and studied more regularly than Roman ones. I think this is because the Roman plays were more culture-specific, best understood by Romans, whereas the Greek tragedies and even comedies had more universal appeal.

Greek theater was the origin of theater. In fact, the origin was speculated upon by those to whom we now attribute the origin of theater, such as Aristotle. Greek theater developed both comedy and tragedy, with a preference for tragedy, the guidelines for which were famously defined by Aristotle. Roman theater was an imitation of Greek theater, but there were some cultural differences. Romans preferred comedies to tragedies. Women were allowed onstage but only in mimes, or plays or sections of plays with no dialogue. One thing Greek and Roman theater shared in common was the wearing of masks. Firstly, they allowed a limited number of actors to play multiple roles that were differentiated by a change in mask. Secondly, masks symbolized types of characters and thus could represent characters who were young, old, happy, sad, and so on.

You can google examples of amphitheaters to see some of the differences. There is a fine example still standing in Epidavros which I had the good fortune to go and visit. You will also want to focus on the role of the Chorus in your response to the question. What is interesting about Greek plays, such as Oedipus Rex, is the way that the Chorus acts like another character that is involved in the action but also separate from it and free to comment upon it. This is an interesting dimension that is largely absent from Roman drama.