The first difference is that there really was never a "Greek Empire." In classical antiquity, the city state of Athens had several colonies scattered around the Mediterranean, and tried to expand its power via the Delian league, but despite the fears of Sparta, the hegemonic ambitions of Athens never really led to the formation of a genuine empire. The eastern part of the Roman Empire evolved into the Greek speaking Byzantine Empire, but this was never actually called the "Greek Empire", as it was a Greek-speaking part of the Roman Empire, that continued in the east after the fall of Rome itself.
The second difference is that an "empire" is a political entity and the Renaissance a period. The term Renaissance refers to the early modern period which saw a "rebirth" of ancient learning in western Europe in the fourteenth through seventeenth centuries. During this period, there were many different empires in western Europe. Like the Roman and Byzantine empires, but not like the fragmented Greek city states, many of the European powers of the Renaissance conquered vast foreign territories. The way in which these conquests differed from the Roman, Byzantine, Arab, and Ottoman conquests was that improvements in maritime technology allowed conquests of distant continents rather than just adjacent countries.