Welcome to this heated debate! (sarcasm) Lots of people tend to feel very passionately about one way or the other about how to indicate that an individual and autism go together. Here’s the gist of it: the majority of autistic adults, especially those who have embraced neurodiversity positivity, prefer to be called autistic (identity first.) The majority of people insisting on “person with autism” (person first) are neurotypical professionals and parents of autistic children.
Why? Well, a while back in the history of autism advocacy those headlining the acceptance efforts took their cue from the Deaf Community.  We don’t call deaf people “people with deafness” do we? Not to my knowledge anyway.
Then there is also the fact that autism is integral to the personality of the person, and part of what makes them who they are. For a much more in-depth and better written expansion of this, ASAN has a good write up here.
Neurotypical people who are not ultimately affected by the choice are insisting on person first language as an attempt to make sure the humanity and dignity of the person is recognized. 1) If you need to use specific language to remember that the person you’re talking to or about is a human being, you’re a terrible person and 2) they are insisting that this language is to respect people while disrespecting them by talking over the objections of the people they’re supposedly respecting. You also tend to hear “person with autism” from the same lips of NT parents who go on rants about how they “love their child but hate autism” or consider themselves “at war with autism.” These people do not accept the fact that autism is integral to their child and equate it with cancer.
Here’s the way I see it: African-Americans tell us not to use the n-word, it’s offensive, and non-African-Americans (with the exception of douchy people) respect that and don’t use that term. The Autistic Community says “call us autistic” and that’s what we should do. There shouldn’t be a debate about this at all.
Now, if an autistic individual tells you they prefer “person with autism” then you should not argue with them; use what they prefer. But for the rest, please default to “autistic person.” Thank you.
 NeuroTribes, Chapter 11, section III, 12th paragraph