You'll have to first define the criteria of impact. The greeks help forward philosophy and govermental ideas. The romans on the otherhand promoted the development of war technology and building methods. Actually, both of them improved the craft of making stone buildings.
That's funny this is what i've been studying in History. Anyways I think that the Greek did because they had like math and science (at least that's what my history book says)and they practiced all that like we do today. They also were like big into art and some of the art we use today proabably came from them.
The Greeks inspired the Romans on many levels. However, without Rome's thirst for expansion these ideas may not have taken root in the majority of Europe. They both had a necessary impact on society, but I guess the Greeks are in many ways the first shackle in a long chain.
The Greeks, because they paved the way for many innovations in various fields we still use to this day, ranging from science, math, the arts, and politics. The Romans, in classic Roman style, took what the Greeks did and threw it together with everything else they conquered and added here and there. The Romans are generally more vividly remembered though.
The thing about the Romans was that they were only successful through the combination of different ideas, for instance in their architecture the have the Greek style pillars with Corinthian decoration, but then they will have some hybrid styles too, like with their boats which had a sense of Greek, Trojan (the Romans are believed to come from Troy) and Egyptian in terms of structure and appearance. I would say that in terms of originality the Greeks had more impact as they effectively allowed the Romans to copy their ideas but the Romans had more impact overall because they were smart in saying "this is what has worked through history, we'll combine it with something that works just as well." Which does not happen enough today as people don't seem to learn from the past as much.
The term "Greek empire" is wrong. There was never such thing. Anyway i have to go with us for obvious and various reasons. Regarding the maths part, actually Greeks took it from Egyptians and Greeks evolved (if that's a an applicable term) it. Actually many things came from Asian countries to Greece like the alphabet.