I'll wade in....I don't think Conan could have handled the Spartans overly long. His natural inquisitivness would have caused him to at least hang around and see what "this" was all about. Beyond that, he would have strongly disdained their ascetic heavily regimented lifestyle. I'd imagine he say," No women in the mess? You drink your wine mixed with water? Are you men or dogs?", and so forth until he killed one or a dozen of them and had to hoof it. As for the Spartans, they might be impressed by his physical dimensions but he would still "only" be a barbarian...uncouth, lawless, faithless and no better than one of their helots...in short NOT a Spartiate.
I'm glad that most of you seem aware that the Spartans were more propaganda than reality. Most of the battles they won by reputation alone and probably during the route when the other phalanx broke is when they really did most of their killing. Interestingly, the Spartans began to drop off their bronze armor and by the Peleponessian War they typically carried the bronze faced hoplon and wore the pylos style helmet (much like a cabacete morion or pikeman's pot from the 16th century) and the ubiquitous red tunic of Lacedamon. Also the Greek city states were experimenting with missile and cavalry troops(peltasts or javeline men). The phalanx was becoming outmoded and ineffective, though the Spartans continued to doggedly use it.
It wasn't until Phillip of Macedonian revitalized it with the sarissa, a lighter shield and supported it with cavalry that it became a viable military formation. The Macedonian phalanx was also somewhat more versatile in that they drilled to change direction and were not as static as the older Greek version. Prior to the refurbishment of the phalanx the idea of "refusing a flank attack" was not even considered. Still, even this phalanx was defeated by the Roman's use of heavy infantry armed with javelins and swords.
I used to be overly intriged with the Spartans and spent alot of study into their culture and military organization. Yes, they were probably the first truly "professional" fighting force in that they devoted their lives to war and no other occupation (really not much different than Gallic noble warriors, Germanic Jarls and knights during the Middle Ages). They had a society that supported them to focus on one thing at the expense of all others. Even their food was bland and tasteless. The old adage" all work and no play makes Johnny a dull boy" certainly applies to them. They were a stagnant and uninspired people and as a result were overthrown and their weakness revealed. I much prefer Skythians to Greeks and so would Conan I think