And I would say the difference between them is that we are privy to what is going on inside Emma's skull, and only see Mrs Elton as an outsider.

Mrs Elton is the character that gives me patience with Emma, makes me aware that Emma's faults are not exactly mortal sins - the exposition of Emma's character reminds me of Hillaire Bellocs eulogy of another spoilt little girl, in that the narrator
"Mentioned her virtues, it is true
But dwelt upon her vices too

Mrs Elton is coarse enough to verbalize sentiments we know Emma holds but does not explicitly state. Emma's sins are more of ommission, Mrs Elton's, of commission. We know of Emma's goodness and sweetness in her treatment of Mr Woodhouse, Harriet, the little Knightleys. And she is no Johnny-come-lately attempting to establish herself in Highbury society.

Holding Mrs Elton up as a mirror for Emma, showing all her worst qualities in frightful caricature, seems to bring out the best in Emma, as well as to remind the reader of Emma's best.
When it comes to doing her duty to the Bates, Mrs Elton has a stronger effect on Emma than all Mr Knightely's advice, and all Jane Fairfax's worthiness.
"it is very shameful. Of the same age, and always knowing her, I ought to have been more her friend. She will never like me now. I have neglected her too long. But I will shew her greater attention than I have done."

Even though Emma has been prompted to alter her ways merely when a state of warfare with one young lady might be supposed to recommend the other, we know (although the good folk of Highbury might not) that Emma blames no-one but herself, sees clearly that her thought, if not her behavior has been self-important, presuming, familiar, ignorant, and ill-bred, and sets about finding a remedy immediately.