Once they did go they marched to Marathon in a record breaking march. They arrived after the battle and congratulated the Athenians on their victory.
The Spartans refused to fight beyond the Pelopponnese at that time because of the phase of the moon. They humiliated themselves in front of the rest of Greece, making the PR victory at Thermopylae a necessity. I’ve wondered if — in line with their later behavior during the Pelopponnesian War — they had taken cash from the Persians to stay out of the fight, as their army arrived at Marathon far too quickly (considering the distance) after the battle.
http://www.livius.org/man-md/marathon/marathon.html
[snip] At the same time, they sent a messenger named Pheidippides to Sparta, who returned three days later (after covering some 450 kilometer!) with the message that the Spartans would send reinforcements as soon as possible. Unfortunately, a religious law forbade any military operation until full moon, which was still six days ahead. (This full moon allows us to date the battle to 10 September or 12 August 490.) [end]