I'm pretty young but I hate the term genderqueer.
Primarily this is because I don't see myself as queer in any sense. I am a pretty straightforward person who happens to define as neither male nor female. I don't want to be part of all that baggage, have bright hair or huge piercings. I don't want to read up on all the latest fashions and trends in political correctness and alphabet soup. The whole term sounds too Shoreditchy, skinny-jeansy, big-pointy-shoesy, lenseless-glassesy, faddy for me to use as my identity.
I do like androgyne, mainly because it was the first word I found, and although the meanings of the root words man-woman, mean that it is not the most accurate description in a totally literal way, when did etymology completely explain the word? I am also comforted by the Greekyness of it, it feels warm and comfortable to me.
As for difference in definition, I have always used androgyne in a few ways. On here it is an umbrella term, I always use it to apply to myself to describe a non-male, non-female person. I use genderqueer to describe an androgyne with a performative/presentational aspect. I often get told of for this though.