Athens is one of the oldest cities in the world and was a center of cultural achievement long before the birth of Christ. It has earned the nicknames "Cradle of Western Civilization" and the "Birthplace of Democracy." It was the site of Plato's Academy and Aristotle's Lyceum. One of the most renowned architectural structures of the ancient world, the Parthenon, still can be visited today; surviving examples of monuments and art can also be seen in modern Athens. The culture of ancient Athens had a great effect upon the later Roman Empire. Ancient Athenian philosophy was based on reason and exploration, and it still influences both modern philosophy and modern science. Homer, Sophocles, Aeschylus, and Aristophanes were some of the greatest early writers, and Sappho was an influential poet. Mathematicians such as Euclid, Pythagoras and Archimedes were among the earliest analysts of their art. Athens also proved to be a hotbed of architecture, astronomy and art.
Describing Sparta's cultural achievements can be tricky, as their culture was not like that of other places. It is also hard to figure out because they didn't write about their own culture - we have to rely on outsiders' perspectives. Their militaristic lifestyle was unique, and it gained them a reputation of being fierce, unbeatable warriors. They valued strength, restraint, austerity, and their religion. Unlike Athens, who had things such as democracy and philosophies which are still used today, Sparta's only real lasting effect was their image or "mirage". Basically, how people perceive Sparta. Sadly, because they didn't write about themselves, we can't know how accurate the information we have on them is, so this question is a bit hard to answer (at least on the Spartan side of things).
The Spartans did not have many cultural or artistic achievements because they focused solely on having a superior military.