The Persian wars were between the Greeks of the Eastern Mediterranean and the Persians. The Persians wanted to stabilise their empire while the Greek cities were rebelling within it or supporting those rebellions from the outside. After 50 years of warfare, the Greeks outlasted the Persians and an agreement for Persia to recognise their independence was arranged in 449 BCE.
The Peloponnesian war was between the empire established by Athens out of the cities which allied to repel the Persians, and a Peloponnesian League headed by Sparta which opposed Athens when it transferred its attentions to its Greek neighbours. It lasted 431-404 BCE, at the end of which Athens lost the empire it had created.