Emma was not personally vain about her looks, and not jealous of other women's good looks, she admired Harriet as a very pretty girl, and although she did not like Jane, Jane's good looks almost softened her when they met after a long absence (instead of increasing her envy or jealousy).

But Emma did care about being accomplished -- e.g. Mr Knightley in chap 5 --

"Emma has been meaning to read more ever since she was twelve years old. I have seen a great many lists of her drawing up at various times of books that she meant to read regularly through ... But I have done with expecting any course of steady reading from Emma. She will never submit to any thing requiring industry and patience, and a subjection of the fancy to the understanding."

That is among Emma's faults, she would not submit to industry and patience. After listening to Jane Fairfax's superior piano playing -- chap 27 --

"She did unfeignedly and unequivocally regret the inferiority of her own playing and singing. She did most heartily grieve over the idleness of her childhood -- and sat down and practised vigorously an hour and a half.

She was then interrupted by Harriet's coming in; and if Harriet's praise could have satisfied her, she might soon have been comforted."

The reasons Mr K considers the intimacy between Emma ands Harriet to be a bad thing for them both -- chap 5 --
"
I think her the very worst sort of companion that Emma could possibly have. She knows nothing herself, and looks upon Emma as knowing every thing. She is a flatterer in all her ways; and so much the worse, because undesigned. Her ignorance is hourly flattery. How can Emma imagine she has any thing to learn herself, while Harriet is presenting such a delightful inferiority? And as for Harriet, I will venture to say that she cannot gain by the acquaintance. Hartfield will only put her out of conceit with all the other places she belongs to. She will grow just refined enough to be uncomfortable with those among whom birth and circumstances have placed her home. I am much mistaken if Emma's doctrines give any strength of mind, or tend at all to make a girl adapt herself rationally to the varieties of her situation in life. They only give a little polish."

My criticisms of Emma is NOT that she is nothing more than a jealous, spoiled brat. She certainly has a very loving heart, and is a most affectionate and dutiful daughter. But she has her faults, and part of it was due to the fact that she was rather spoiled, and her family was rich and of great consequence in the neighbourhood. She is kind and generous to the really poor, but conceited and snobbish to those slightly below her e.g. the Martins and the Coles. She said of the Martins -- chap 4 --

"The yeomanry are precisely the order of people with whom I feel I can have nothing to do. A degree or two lower ... I might hope to be useful to their families ... But a farmer can need none of my help, and is therefore in one sense as much above my notice as in every other he is below it."

The Coles were "low of origin, in trade" and Emma had intended to snub them by refusing theri dinner invitation -- chap 25 --

"The Coles were very respectable in their way, but they ought to be taught that it was not for them to arrange the terms on which the superior families would visit them. This lesson, she very much feared, they would receive only from herself;". But then she found she did not like to be left in "solitary grandeur", and "did not repent her condescension in going to the Coles" (chap 27). Condescension! (the word most often used to describe Lady Cat in P&P). I think Emma, like Darcy, had been "given good principles", but was also spoiled and "left to follow them in pride and conceit". Darcy had to meet Elizabeth before realizing that, while Emma had Mr Knightley.