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Hmmm, good question.
With Kazakiri, well, it was mainly due to the fact that he and Index befriended her; as she was an AIM thought being. She didn't exist on the same plain as everyone else, so when the opportunity arose for her to end that solitude, she took it. Touma influenced her because he did what he always had done; he helped her and viewed her as a person and not anything more or less than that. He had no reason to because, in his mind, she was yet another person who he needed to save.
With Misha...that I don't know. She's a mystery to me even now, and her motivation behind her actions in the LN was mainly based around feeling obligated to help Touma because they were working to achieve the same goal and that was to stop Fiamma. With her, I'd say it was more like mutual agreement to resolve a bad situation more so than anything else.
This is just what I believe, so I'm not sure if it's completely accurate.
As for what makes them different; one is in this current plain of existance and actually is composed of flesh and blood, while the other is not.
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