I dislike the word "Asperger's" for several reasons, partly because I find it hard to pronounce. If I'm talking to somebody in real life I say "Asperger syndrome" (I prefer it without the "s" on the end) because I know they will know what "Asperger" means. More people have heard of it, in my experience.
If I'm talking online, however, I prefer to use the the term "Autism Spectrum Disorder" or "autism spectrum disorder". Even though it says autism spectrum disorder on my diagnosis, I never feel like I deserve to use the term Autism Spectrum Disorder because the word "autism" implies a certain degree of severity.