From my Aspie perspective, I much prefer listening on headphones because then I'm not infringing on anybody else's sound space, and it makes it easier for me to listen to the music and ignore (many) interruptions. It also means that nobody else is going to be listening to the music I like and making negative comments about it.
I also have a home recording studio where I record a lot of my own music, and that's pretty much all done on headphones. So I like listening to "properly mixed" music on the phones, just so that I'll know what to aim for when I'm recording and mixing.
When I do listen to music on speakers, even in the car, I tend to keep the volume very low. And it bothers me when I'm trying to concentrate on a project and somebody else's music is distracting me.
But I do understand how some Aspies may feel it's really important to hear the music in the best possible way, and prefer speakers, or to have our own preferred music on while we're doing housework, driving around, etc. Sometimes speakers are what work, but I prefer headphones.