This seems like such an obscure word, barely mentioned on internet searches & definitions seem to just sum it up to be an identical synonym to immortality.
Side not: vim's spellchecker does not even register athanasia as a real word.
So, I'm going to narrow down the definition to what I believe the word means. I think I got the feel of the narrowed down definition from one of CGP-Grey's podcasts on immortality, where he mentioned that he wants to be immortal and young, which he described as athanasia (I think).
So that's what I believe it means. athanasia is a subset of immortality where you stay functionally young.