Did the differences between Athens and Sparta lead to the Peloponnesian War?
History shows us that rival powers that are relatively evenly matched will tend to find reasons to go to war with one another. They may talk about things like political views as their causes for going to war, but what is really going on is that each side simply wants more power. This is what was going on between Sparta and Athens in the Peloponnesian War.
There were clearly differences between Athens and Sparta. Athens was a mercantile, relatively democratic state. Its people were mainly concerned with the good life -- with having material goods, art, and political rights. By contrast, Sparta was a "garrison state" where an elite trained itself for war while subsisting on the labor of a mass of people who were essentially slaves.
However, this is not what made the war happen. The war happened because Athens (after the defeat of the Persians) tried to expand its power using its naval forces. This frightened the Spartans -- they feared that this power might some day be used against them. This was the true cause of the war -- Sparta's fear of Athens' power.
While there were serious differences between the two poleis, these differences did not really cause the war. The war was a struggle for power.