Most of the Greeks taught logic, rhetoric, and grammar--the three becoming the central core of most Greek education. Socrates was a product of this liberal education, and was the creator of what is now known as the "socratic method." Plato, a student of Socrates, further developed education in ancient Greece, by his methods of creating dialogues and by arguing for a type of education that matched or befitted a person's class and role in society. Finally, Aristotle believed in the rational nature of the human being and saw the world as being ruled by natural laws.
Education, or "pedagogy," takes many diverse forms, although we all owe a debt to the ancient Greek educational development. Today the educational "buzz-word" is "differentiated instruction." What this means is education using as many diverse and engaging approaches and perspectives as possible. Whereas before education was about a teacher "lecturing" in typically Greek fashion to a collection of gathered students, now we have students doing the learning, teachers using innovative technology, and even cyber classrooms--so mainly the difference is in the type of technology we use to get the learning objectives done.
A lot of things can be said, but I will focus on two important aspects of Greek education. First, we cannot underestimate the importance of Homer's Illiad and Odyssey in Greek education. They studied these writing with great precision and probably committed much to memory. In the sands of Egypt, at times we still find papyri of Homer, which were probably the work of school children! When we compare this to us, there is no one text that is as important as this one. Second, rhetoric was an important part of their education, whereas it plays almost no role in our education system. We like to concentrate on content and very little on presentation. This is unfortunate, in my opinion, because our content is somewhat lacking. Also rhetoric can be important as well. These two points should underline some of the differences.