Don't know why people are so absolute on Achilles winning, when the stories about his flawed immortality are contested by different sources (some say arrows drew blood from his other limbs and so on). Leonidas was just a man who had no god-given traits, aside from his distant relation to Hercules. If you're going to give Leonidas any special abilities at all, it would be slightly better strength than those of his peers. That being said, I can see the battle going both ways. Leonidas is a good tactician, and is as one should be on the battlefield; fast and strong. But that's all he has going for him. He isn't superhuman, and he has no godly traits as Achilles has (for the sake of the fight, we'll assume that Achilles has his infamous weakness at the heel, and is borderline impervious to anything Leonidas can dole out). The battle will definitely be decided on whether or not Leonidas can deduce Achilles' weakness and strike at it. If it is, the battle is as good as over. Adversely, Achilles is a fighter as well, and I can see scenarios where Leonidas simply cannot find out Achilles' weakness.
Thought this was a Fate thread and came in to say Achilles fucks him up. My answer is still the same tho. Leonidas has no way to deduce Achilles' weakness despite him being a good tactician. There's no reason for him to assume there's a weak point and there's no reason for him to notice that it's exactly the heel. He'd be pressured by Achilles' skill as it already is, so immortality just tips everything into Achilles' favour.