Peloponnesian War.
Athens and Sparta did not fight each other in isolation. They fought as allies and as adversaries in various conflicts over a couple of hundred years. So Athens did not defeat Sparta or Sparta defeat Athens. They were involved in wider conflicts in the battles.
Sparta and its allies and Athens and its allies fought for domiance. The Spartan side won with Persian assistance.
There were not 36 wars between Athens and Sparta.
In Sparta you had to join the army or navy when you were 7 years old! But in Athens you had to go when you were 16. Another difference is that if you where a girl in Sparta you had rights. But in Athens you barley have any rights.
Sparta was an aristocracy that had a Council of Elders. Athens was a democracy. It had a council too, of 500 men over 30 years old.
Athens was one of the 2,000 city-states which fought each other over a thousand years.
The Peloponnesian War was between Athens and Sparta. It was after the great Persian war, when Athens and Sparta worked together to defeat Persia. The Peloponnesian War was started because Sparta feared Athens may dominate them one day. There was a big battle between them, lasting for many years, but in the end, Sparta won.
The Peloponnesian war was Between Athens and Sparta. Sparta was jealous of Athens and hated their idea of democracy. Sparta was strong on land, Athens at sea. But in the end, Sparta won and established an Oligarchy called the 30 tyrants. But 50 years later when Sparta was still weak from fighting Athens, a kingdom to the north called Macedonia, led by Alexander the Great, conquered Sparta.
It involved the Greek in a destructive struggle for ascendancy between the Athenian empire, and the Peloponnesian league led by Sparta. It led to the dissolution of the Athenian empire, massive destruction in the Greek world, and solved little, with inter-city warfare continuing on amongst realigned allies afterwards.
After 27 years Athens, its fleet destroyed, was besieged and starved into surrender.
Sparta and Athens were at various times allies and adversaries. The main clash was the Peloponnesian War 431-404 BCE when the Spartan Peloponnesian League fought Athens and its empire in a devastating war whic embroiled the Greek world from Sicily through to ASavesia Minor.
Sparta had a warrior society and the Athens had a a democratic government. Sparta boys began training a 7 years old for a lifetime. Athenian boys attended school if their families could afford it.