For example the requirements for the most common way to earn the GM title include a requirement that your rating has reached 2500. The live rating is used for that.
I always believed it is because either FIDE ratings are only updated once a month (see 7.1[1]) or because their ELO is too close, therefore the prob. of winning is 0.5 for both anyways.
[1] http://www.fide.com/component/handbook/?id=172&view=article
I heard (possibly from the Chess24 commentators) that if their difference in rating was 4 or more, each draw would bring them closer together in live rating - Magnus would lose rating with each draw; but as it's 3, their ratings don't change. i.e. anacleto was right, thomasahle wrong.
Also, Elo isn't an acronym (ELO) as one might think - it's named after its inventor Arpad Elo. https://en.wikipedia.org/wiki/Elo_rating_system
Interestingly, when I first looked at this page, most of the good/correct comments were voted down, most of the bad ones not, it was weird. (It made me suspect that other HN pages are like that but I don't know it! - I know chess better than most subjects discussed on here) But that has now almost totally been corrected.