This is a complex question and one that is still debated today. In light of this, I think we can answer this question in two ways. On the one hand, we can say that the war was started due to pride and jealousy. This might seem like a foolish answer, but it is the answer that the great historian Thucydides gave. Not all wars are started because of economic motives. Pride plays a role as well. Here is a quote from Thucydides:
"The real cause I consider to be the one which was formally most kept out of sight. The growth of the power of Athens, and the alarm which this inspired in Lacedaemon, made war inevitable."
We can also answer this of causes from a different perspective. We can say that the tipping point of war came when the Athenians put an peacetime embargo on Megara. Corinth at this point got other city states to petition the Spartans to attack Athens and they did.
This was an extremely long war, which almost destroyed the city states of Greece. In the end, not much was accomplished and Sparta won the war through the generalship of Lysander. Athens surrendered.
In terms of how the war shaped philosophy, it is hard to say. However, a lot philosophy at this point discussed what was the best form of government.