Some of this sounds familiar to me as well, in that I have a younger child who connects much more effortlessly with us emotionally (in particular, he knows when to offer an apology, when to back off, and when to offer words of concern--all of which DD is not so great at). However, my older DD is popular at school and has been (casually) assessed as not having ASD by several professionals. She is different than your DD in that she's very sensitive; in fact, she picks up on my moods and expressions a little too well, but does not respond the way her brother does (this can make it hard to feel positive towards her). I actually do think my younger child is unusually socially acute--he is also extremely positive-natured-- so the difference is especially noticeable.
There is also something to be said for the concept of child-parent fit. This can be very hard to accept. I still struggle with it, because it's a taboo subject. In life, some realtionships are easier. We can work on the harder ones. They may always be harder, though.
I do second the book PLayful Parenting.
