Nesta has always had a much closer relationship with Elain than she has with Feyre. Feyre was never particularly "doted upon"- in fact, Elain was- so it's not out of jealously. But the reason is still blatantly obvious.
No two characters (with debate, of course) are more alike than Feyre and Nesta. Yet Feyre was always the hero- the one that kept the family alive. She was the good one, and no one can say that's not a viable reason for Nesta to hate her.
Not when Feyre is everything that Nesta is- and everything Nesta wishes she could be.