You know, honestly, since I've started my internship working with kids on the spectrum, I've had to question this myself.
I typically have no problem with empathy. I relate well to others in that regard (human and animal), and indeed one of the reasons I opted for this field is because growing up I was always THE person my friends came to with problems - from your typical teenager relationship issues, to "my stepdad has sexually abused me", to "so-and-so on our track team tried to rape me".
But my more general social skills SUCK. I was always the kid in the back of the class, reading a book, speaking only when spoken to. Virtually every single friend I've made since I was 5, was me sitting around looking awkward, and someone else deciding to come up and start talking to me. I came out of my shell quite a bit in college and thereafter, but the friends I made still were initially the ones to actually start talking to me, rather than vice versa. I seem to have retreated a little back into my shell since undergrad - life circumstances that happened right after college took a blow to my self-esteem (and it took me most of my college years to build any self-esteem, as growing up it was virtually nonexistent), plus being a single adult who isn't into partying, or clubs, or bars, have somewhat shortened my regular social circle. So when I first started working with the ASD kids, it amazed me how much I seemed to have in common with them, and how much it felt like, while I was helping to teach them the various social skills, I was teaching myself as well. So while I don't feel like I'm on the spectrum (I feel, at this point, that a lot of my social skills problems are/were situationally/environmentally based - bullied at school, "difficult" home life, etc.), I definitely see some things I seem to have in common with many that are.