I see Elizabeth as a person who only values or treats other people well as they are of use to her. Mrs. Clay knows all the right things to do and say and so she is of material use to Elizabeth's ego. Anne won't play that game, so Anne is nothing.

Anne couldn't (at least not up until now) introduce Elizabeth and Sir Walter to anyone 'worth' knowing, she adds no consequence to their social circle or status, so she is nobody. There are others who are considered 'nobody' by Elizabeth and Sir Walter, including Frederick's clergyman brother--they just don't have the misfortune of having to live with them.

Also, Elizabeth is a very unhappy person, and Anne is a convenient target.

(Ch. 1) "Thirteen winters' revolving frosts had seen her opening every ball of credit which a scanty neighbourhood afforded; and thirteen springs shewn their blossoms, as she travelled up to London with her father, for a few weeks annual enjoyment of the great world. She had the remembrance of all this, she had the consciousness of being nine-and-twenty, to give her some regrets and some apprehensions."

Elizabeth feels that her life is passing her by, much as Anne has been feeling, but Elizabeth has no inner resources to deal with this, so she lashes out at the most convenient target. Also, Elizabeth has had no guidance from a parent since the age of 16 to tell her that this behaviour was wrong. She won't listen to Lady Russell, and who else is there?