I do like the description of Mrs Elton as a “frightful caricature” of Emma, and think it sums up the relationship between the two characters very well. (By this I don't mean the way Emma and Mrs Elton interact, but rather their functional – or is it structural? - relationship in the novel as a whole. Perhaps if I had studied Literature rather than Sciences I would be better able to explain what I mean!)

As you say, we see Emma's capacity for kindness and patience, for example with mr Woodhouse and with Harriet. We are also shown her thoughts and feelings. We see her conscience in action, her shame at her errors and her innate honesty with herself.

We cannot 'hear' Mrs Elton's internal dialogue, but only see her from outside. Much of what we see is a woman who is endlessly trying too hard to impress, and working to boost her own image. This leads me to think that much of her behaviour is rooted in social anxiety and the fear of being judged socially inferior because of her 'low' connections and background.

Like it or not, egalitarian ideals were not popular in England at that time, and in the aftermath of the French revolution were viewed as dangerously radical. English society was acutely conscious of nuanced gradations of social rank based on birth and wealth. I think this underlies an important distinction between Mrs Elton and Emma.

Mrs Elton is anxiously clawing her way up the social greasy pole, and is constantly getting it horribly wrong in her efforts to appear genteel. In contrast, Emma simply doesn't feel the need to try in this way. Her social position at the top of the Highbury social heap is assured by virtue of her wealth and her birth into the younger branch of a very ancient family.