All we know about the Spartan case comes from a passing remark in Herodotus 7.206.
The Spartans sent Leonidas and his men on ahead in the hope that the sight would inspire the rest of the allies to arms, and discourage them from joining the ranks of those who were already collaborating with the enemy, as they might if they got the idea that the Spartans were holding back. Later - that is, after celebrating the festival of Carnea, which was holding them up - they planned to waste no time in sending every available man to join Leonidas...
The Olympia festival took place around the same time - if both festivals finished at the full moon, then exactly at the same time, though the Carnea was longer and would have started five days earlier.
There are a few plausible interpretations of this.
A widely-held view among historians of the period is that Herodotus's explanation is wrong. The Olympia festival did not stop other Greek cities from sending over 50,000 men and 271 triremes to Artemisium, a battle that took place at the same time. It is difficult to believe that the Carnea was a special case.
Having said that, there are other incidents of Greek city-states in that era delaying or refusing to send a military force because of religious festivals, including sometimes sending military forces selectively (Sparta delayed liberating Chios from Athenian occupation in 425/4 because of a minor festival, but they did invade Attica at the same time). So it's not intrinsically implausible.
Exactly what kind of festival the Carnea was is not clear: one point of view holds that it was a harvest festival, another is that it was a military initiation festival. If the latter was the case, that would provide a solid practical reason for delaying: they may have been waiting for a large fresh batch of recruits.
In Herodotus' report, the bit about "discourage [other Greeks] from joining the ranks of those who were already collaborating with the enemy" implicates Argos. Argos had been Sparta's rival for hegemony over the Peloponnese throughout the previous century; Argos was a Dorian state like Sparta, and so also celebrated the Carnea; Argos remained neutral in the Second Persian War; and an infantry force going from Sparta to Thermopylae had to pass through Argive territory. Neglecting the Carnea could well have given Argos licence to do more than sit out the war, and actively fight on the Persian side.