The ancient Greeks were the birth of the western world, they invented western philosophy and were advanced in both religion, society and military tactics. But their advances were also plagued by war, infighting and chaos, many states within greece would constantly fight each other. And although they were extremely influential in terms of power they were dwarfed by competing empires at the time like the Persian empire.
The Romans on the other hand had perfected a modernized society within the ancient world, conquering most of it. During its extremely long reign I think its hard to argue that the Roman empire wasn't top dog. I'd have to say Roman just because, although the Greeks were cool Rome just was much cooler.