Has Greece left more of a cultural impact on the modern world than Rome?

Athenian democracy developed around the sixth century BC in the Greek city-state (known as a polis) of Athens, comprising the city of Athens and the surrounding territory of Attica. Athenian democracy is often described as the first known democracy in the world. Other Greek cities set up democracies, most following the Athenian model.

Ancient Greece left a cultural impact on Rome lol. Their writings are directly influenced from theirs. So when you speak of their influence on Modern World the answer isn’t simply this or that.

They both contributed plenty. Picking one more so than the other can be attributed to how one aligns their priorities and the weight of importance they assign to each. Try it ... it's a learning experience. This is what historians do when they debate conjecture. In other words, it's close enough where it's a matter of opinion. I can easily make a case for either but I can't decide your opinion for you.