This conversation may very well change from language to language, but - particularly in English - the Person/Identity-first rhetoric is largely pedantic and causes turmoil where none is needed. Both 'sides' often claim their view to be superior, and many even get offended if you use the other way, whatever 'side' one may be on. This is primarily an issue in the autistic community, but it can and does extend broadly to other uses where a qualifier might be used to identify a person, such as bipolar, disabled, and possibly sexual orientations, ethnicities, or anything else someone can find a way to get offended based on how one chooses to word what is basically the same sentence.
This is how it works: 'person-first' language would be "a person with autism, they have autism," while 'identity-first' language would be "an autistic person, he is autistic." The idea is that person-first rhetoric is supposed to avoid defining a person with a label, while identity-first rhetoric is to avoid dismissing a pivotal aspect of an individual's identity. This is what they're supposed to do, in the eyes of those who make such distinctions.
I'm perfectly fine with people having their own preference, particularly when referring to themselves, but when you push either on someone as being the 'correct' or 'superior' way, you're causing unnecessary turmoil and potential conflict that never needed to be there, in the first place. It's largely a distinction without a difference: pedantic. The primary difference is between the usage of the verbs (and their different forms) 'to be' and 'have.' One can be autistic, or one can have autism. The primary difference is that one is a noun and the other an adjective. It's a bit misleading to say someone is autism, and it simply makes no sense to say someone has autistic. Therefore, grammatically, it is the semantics which determines which verb you use.
People who identify with the diagnosis of 'Asperger's' can find themselves in a bit of a pickle, regarding this distinction. On the one hand, it's grammatically proper to say someone has Asperger's, but it doesn't make any sense to say someone is Asperger's. There is a term, 'Aspie,' which can be used in lieu of a better term for this, leading to sentences like "she is an Aspie," but it's inaccurate to say someone is aspergery, which isn't even an official word, but primarily a slur used to describe people perceived as having Asperger's-like stereotypes (socially awkward, nerdy, even as an alternative form of 'dumb' or 'retarded.') But 'Aspie' can, in some contexts, also be used adjectivally, such as, "I feel an aspie meltdown, coming on..." People, however, are seldom describes as being aspie. It's not out of the question, but it's rare.
In conclusion, the distinction is primarily of personal preference, while pushing any as truly 'superior' is merely pedantic. I was once verbally attacked by someone for calling myself an 'Aspie,' because they felt it was apparently offensive. There are people, both within and without the autism community (though they seem few and far between, in all) who consider 'Aspie' a slur, despite having both it's origins and propagation of use centred in the heart of the autism community. That person thought they were defending the honor of those diagnosed with Asperger's by and large, while simultaneously attacking any who self-identified as 'Aspie,' which is a bit like saying, "How dare you call someone gay! They're obviously a homosexual." While 'a homosexual' certainly isn't the most endearing phrasing, neither is really incorrect.
Learn what others prefer to be called, and try to call them that. Identify what you prefer and call yourself, and call yourself that.
