The Athenians were arrogant in their wealth and were focused on art and philosophy, but had a powerful military.
The Spartans were focused on their military forces. They didn't like the arrogance and balance of power of the Athenians. The differences in the 2 city-states is what started to trigger the Spartans to fight.
The Peloponnesian War
The Spartans marched into Athenian Territory.
Sparta and Athens fought for 9 years
The Spartans defeated the Athens while they decide to surrender.
Both Sparta and Athens signed the Thirty Years' Peace Treaty.