Thucydides, who was there at the time of the Peloponnesian War, said that the war was inevitable because of “the growth of Athenian power and the fear which this caused in Sparta.” While some scholars disagree with Thucydides, this seems like as good an explanation as any for why Sparta and Athens went to war.
Sparta and Athens were rivals for power in the Greek world. Each of these two poleis was powerful and aspired to lead the Greek world. The two poleis had very different societies and cultures. In a sense, you can compare this to the situation between the US and the USSR during the Cold War. These were rivals who were vying for power and influence.
Eventually, Athens went too far for Sparta and its allies. Athens had created the Delian League to help Greece fight the Persians. However, Athens then made peace with Persia so it could turn its power to the task of dominating Greece. This worried other poleis, as Athens tried to maintain power over the Delian League even as some of its allies tried to leave because the reason for the league was gone. As Athens tried to keep its client states in line and to expand its power in Greece, Sparta and other poleis such as Corinth became more and more concerned. Eventually, the two went to war. This was a case in which competition for power eventually led to a war between two great rivals.