the Greeks at Thermopylae (not all were Spartan) killed both slaves and Immortals. they were eventually beaten when the Greek flank guards deserted and the Persians were able to surround them. the Immortals did not play a distinguished role in this action. also, the number 300 can be regarded as a wild guess and totally unreliable. the other numbers Herodotus (the historian from whose writings the battle has been reconstructed) gives, like two million Persians, are wildly implausible.
actually no, the immortals were sent in during the fight and were devastated. its also not widely known that while 300 spartans stood there, there were also about 2200 other greek soldiers who really did the bulk of the fighting. the spartans are famous because when the army was outflanked, they stayed behind to cover the retreat of the other greeks. to answer your question, its simple. if the immortals attack head on, as they did in the middle of thermopylae, they would be destroyed, as they were. if they attack the spartans from behind, as the did in the end of the battle, the greeks are doomed... as they were.
Didn't the Spartans manhandle the Immortals at Thermopylae? Wasn't this decided like 2500 years ago?
Spartans, no doubt.