I think there are elements of both dislike and jealousy in Elizabeth's psyche with regard to Anne. Without the slightest desire to win the regard of either CW or CM, Elizabeth, having already seen the youngest sister marry and with full awareness of time passing, could very easily resent the fact that Anne has actually had 2 proposals of marriage to her 0. It is clear that she just plain doesn't like Anne (and complex matters relating to the late Lady E may be at play here; we know nothing of Elizabeth's relationship with her mother). Anne compounds the issue by being quietly independent, and having a social life (thanks to Mrs Smith, the Crofts, the Musgroves and Lady R). Even though it is highly unlikely that Elizabeth would feel herself overmatched by marriage to a Duke, the fact remains she has never been asked, and she knows it. It must be galling to see someone as disliked as Anne being popular at any level, having proposals of marriage, and generally being quietly busy and content without reliance on or concern with Elizabeth. For someone who wants to feel important in all spheres of influence open to her, Elizabeth really has little actual importance to anyone but herself and her father and (of course) Mrs. Clay. No wonder she waxes mean and malicious when she can.