How might the Persian War have ended differently if the Spartans had not held out at Thermopylae

The battle at Thermopylai did not alter the course of the war at all. It was staged to force a sea battle in the strait next to Thermopylai where they hoped to destroy the superior Persian fleet in the narrow waters. This objective was to remove the amphibious threat posed by the Persian fleet to the Greek cities, which kept their armies home to defend them, and allowed the Persian army to pick them off one by one. It also protected the Persian supply fleet, without which the Persians could not sustain themselves in such a poor country as Greece.

The sea battle continued for the three days that the pass of Thermopylai was held by the 5,000 men of the Greek cities including the small Spartan contingent. Unfortunately for the Greeks, they had the worst of the sea battles and had to retire. So the Thermoplai engagement was to no avail.

However the Greeks later reenacted their plan in the sea battle at Salamis and destroyed the Persian sea power. Half of the Persian army had to go back to Asia as it could not be supplied, and with the amphibious threat to their homelands gone, the Greek cities concentrated their forces at Plataia the following spring and defeated the remaining half of the Persian army and its Greek allies.