It's the difference between being born to wealth and making your own fortune.
Rachel has power from her heritage and her access to the Phoenix, but she's rarely had to WORK to gain power, whereas Emma has.
Rachel's anger is only weirdly understandable from her POV as a time displaced person from a no longer valid timeline. All parties involved were adults, it literally was nothing to DO with her, and she's acting very, very childishly about it; "You took my daddy (who isn't) from my mommy (who wasn't) and so I'm going to pull your hair"
Oh, and Kitty never trusted Emma, that's why Emma asked her to join the Astonishing X-Men.