Why did the Spartans and Athenians unite during the Persian Wars?

Sparta and Athens were natural allies who provided support to each other - Sparta leading the Dorian bloc of independent city-states, Athens the Ionian bloc, with both interested in maintaining peace.

So when a Persian expedition was sent to subdue Athens in 490 BCE after its interference in the Ionian Revolt against Persia, Athens naturally asked Sparta for support (however Athens had defeated the Persians at Marathon by the time the Spartans arrived). When Persia sent a full scale invasion in 480 BCE, the southern Greek city-states united to repel it under the leadership of Sparta with Athens providing the largest naval component.

They remained friends and allies until 460 BCE, when Athens sent an army to help the Spartans put down a revolt of its serf population. During this campaign they fell out and avoided each other. After the Persians agreed to peace the Athenians turned the Delian League into an empire of its own, became over-adventurous and clashed openly with the Peloponnesian League led by Sparta, resulting in the 27-year Peloponnesian War which Athens lost.