Everyone expects Emma to be the bigger peson because she occupies the supeior status -- both Emma and Jane were gentlemen's daughters, but Emma was rich and Jane was poor, it was for Emma the rich gentlewoman to do more to extend the hand of friendship to Jane the poor gentlewoman.

And in their mutual apologies at the end -- Emma was actually conscious of earlier neglect as well as the wrong she had more recently done in the Dixon suspicion, but Jane's apology was for her recent "cold and artifcial" manners when she always had to play a part since her secret engagement with Frank.

I don't think there had been any wrong-doing on Jane's part earlier, Jane's refusal of Emma's kindness was only when she was ill (and unable to think properly) after breaking off her engagement, when she must have believed that Emma was the rival that led Frank to regret his engagement to her after his words on Box Hill.