In a sense, the kinds of tensions between countries that lead to big wars like these are typically caused by the ways in which the countries see themselves and their opponents. Typically, the two sides think of themselves as fundamentally different from one another. We can see this in the Cold War of the second half of the 20th century. During that conflict, the West and the Communist Bloc came to think that they were so different that they could never coexist with one another. The conflict between the Athenians and the Spartans was similar.
Basically, the two poleis felt that their cultures were simply too different. The Spartans valued things like order and stability. They wanted all their people to conform to a certain standard so that their society could remain strong and unchanging. The Athenians, by contrast, valued freedom. They did not feel that everyone needed to conform. Relatedly, the Athenians had a democratic form of government while the Spartans had a monarchy. What happened, then, is that the two sides came to have such different ways of looking at the world that they each felt they could not coexist with the other. This was the real root cause behind the rivalry between these two poleis.