I'll open this post with a question: what is the difference between I have and I am? I guess one could start from looking at a grammatical perspective where one indicates possession (and uses the accusative case or direct object) whereas the other is a state of being; there is no sense of supplementary or additional nature to what is already present, rather denoting a sense of completeness in what is present. Both cases can have differences for a before and an after. Linking back to possessions, 'I am' is not generally used with objects (although I am sure that special situations exist); for example 'I have a chair' is generally heard of more than 'I am a chair' (however I am chair-shaped may be used if you have been sat on a train or at a desk in the library too long, hinting that you're a bit stiff).
What relevance does pondering grammar have to do with autism? It actually has quite a bit.
Firstly there is the use of identity first language vs person first language: to simplify - autistic person v.s. person with autism. When you look at the sentence grammatically it can appear that you have merely changed the noun of autism in the person first example to the adjective of autistic in the identity first example. This is something we did a lot studying languages at school and it university; a good trick to know if you need variety and also to grow your comprehension. So, do both of these examples mean exactly the same thing if there has merely been a transformation of a noun to an adjective?
You would think it would be exactly the same, but in all honesty it is a bit more complicated than that. This small grammatical transformation actually signifies a big difference between the two examples: 'person with autism' and 'autistic person'. Referring back to the original differences between 'I have' and 'I am', the main difference is of a supplementary nature or addition. Autism is not in addition to who someone is, rather it is the way they experience and perceive the world, warts and all. The use of 'I am' makes it personal and does not compartmentalise autism - that is to say it does not separate it off in a box. Much like other medical diagnoses it does affect a lot, if not everything we do, but the sense of lived experience is so rich that it is the way we perceive the world, whether positive or negative.
Let us also not get mixed the concept of 'labelling' and 'acceptance of a diagnosis'. This can be seen as a pitfall of saying 'I am autistic'; some may see this as labelling and giving an excuse to everything that is done or happens with a label. In fact, this is not the intention of identity first language at all. As the phrase 'identity first' hints at, it describes an identity, not a list of symptoms. This can be seen as a thin line to tread, but we all need to be aware of the reasons identity first language may be used. It can show an acceptance of who we are, rather than perhaps the thought of limiting ourselves by imposing a label. Many on the autistic spectrum, whether officially diagnosed or not, often carry long journeys including some of the following: feeling alienated, bullying, being ostracised, feeling misunderstood, confusion... the list is not exhaustive by any means. But through finding out part of our identity allows us to feel more us, whatever we choose to call ourselves. This is really important.
Others, including families or significant others, may prefer 'person with autism', as they may wish to not define their loved one or themselves with autism, or may feel as an individual that autism does disable them. At the end of the day, it is the individual's choice what language they prefer to be used, how they are called and how open they are. Others need to list and follow what their loved ones prefer and be aware of the complex nature of this topic. There is no right and wrong that encompasses everyone, but there is the recognition that we need to listen to individuals and listen to what they wish to be called. There does appear to be a trend for many autistic individuals to prefer ' autistic person' to 'person with autism', others however prefer the latter and others again are not bothered. This highlights how important it is to follow the lead of the individual and respect their decision, whatever it may be. This in part is similar to listening to and respecting people's choice of pronouns; a question of identity which needs to be respected.