IANAG (I Am Not A Greek), nor do I speak/read the language, but I've been going through a lot of greek words the past six months because of my Light Against the Dark campaign. So take this with a grain of salt, but not too big of one.
Athanatos means "undying" or, rather, something which by its nature is incapable of becoming dead. It can also be used to refer to something which is everlasting. That is, it doesn't mean "immortality" so much as it does "an immortal".
The Persian troops fought by the Spartans were also called athanatos in some texts, because (IIRC) as one would die another would step smoothly into place, making the troop seem less like a mass of humans and more like a hydra.
Immortality as a state (rather than immortal as an object) is, I believe, athanasia.