"Why she did not like Jane Fairfax might be a difficult question to answer; Mr. Knightley had once told her it was because she saw in her the really accomplished young woman, which she wanted to be thought herself; and though the accusation had been eagerly refuted at the time, there were moments of self-examination in which her conscience could not quite acquit her." (Ch. 20) made

Well, Mr Knightley doesn't always hit the truth with Emma. His heart is righter than his mind here, I find.
May I tell what I felt, backing in my infancy thoughts, when I was "promising" to become like Emma, for similar reasons?
Well, the acquaintance with a perfectly bred, perfectly accompished creature didn't raise jealousy, merely dullness. Exactly as Emma drew up good lists of books to be read, she would have liked to make her duty of good little girl learning of the example, but it was as boring as reading a series of books well listed, or practising the pianoforte : no delight, above all no surprise (I remember, later, having passed several hours per day at piano because I had loved such piece and wanted to play it; but one can fancy Emma had not had such pieces). An acquaintance with Jane Fairfax, because Jane Fairfax had been said to be so accomplished (by her mother, later by Miss Taylor, before Mr Knightley) was "requiring industry and patience" — chapter 5.
Jealousy comes only from Mr Knightley's praise of Miss Fairfax, because I suspect that she wanted already to be "first" to him, not loving him yet as an adult could, but already feeling for him. And Mr Knightley's enthusiasm to Jane Fairfax (remember when he declares his love to Emma, chapter 49, he nearly speaks more of Jane Fairfax! — like Mr Dixon would make play the same Jane Fairfax instead of his own future wife) was very proper, until Emma understood the depth and exclusiveness of his love to her, to produce jealousy. I think that Mr Knightley not only admires Jane Fairfax, but also aknowledges her as his peer — like Emma aknowledges Frank Churchill as her peer, in chapter 54. And knows that she is not the kind of wife he needs : too perfect, precisely. He needed an Emma...