Athens was a highly artistic city-state. Arts education was valued. Athenians were artisians, merchants, philosophers, warriors, but mostly citizens. (There was a slave system, but it wasn't as intense and degrading as the Spartan slave system.) Athens gave rise to some very important philosophers, such as Plato and Aristotle. It also led to some revolutionary political ideas--democracy and communism.
The Athenians also created the Delian League, a confederacy of city-states allied together to protect one another. The Peloponessian League was created to counter it. The formation of these leagues led to the "Suicide of Greece". War broke out between the two leagues in 431 BC and ended with the Athenians defeated. But, after this, it was easy for the Macedonians to take control.
Sparta was a harsh warrior society. Spartans were very dependent on one another in battle (the
hoplites, or foot soldiers did battle in
phalanxes, which required extreme cooperation and trust). This led to an obsession with equality. They called each other "homoroi", or equals. (Of course, slaves were not equal, but, of course, slaves were a sub-species.) Even women had more equal rights in Sparta than in most other places--Spartan life was war-oriented, and, while the men were off fighting battles, the women had to defend Sparta (often against the slaves).
The Peloponessian League was headed by the Spartans. The Peloponessian League and the Delian League engaged in a war (starting in 431 BC) that would later be known as the "Suicide of Greece". Though the Spartans won, outsiders would quickly control Greece. The Greek civilization was conquered by Philip II of Macedonia.
It is important to note that, while Sparta and Athens had a fierce rivalry, they also banded together to fight the Persians.
This time period was also called the Helenic Ages. It morphed into the Roman Era when Rome took control.