Some people do suffer psychological damage - I have friends with PTSD due to all manner of fucked up shit that has happened to them. Some of them also suffer from severe depression (not the get some exercise and meditate type- not that these things don't help, to a degree - but of the ongoing and pervasive type that renders them deeply unhappy). I think this is a normal reaction to the myriad traumas we might face, given the shit we are born in to. Whether it is about abuse, oppression, gender (I know, criticize me now for identity politics!) or just that the world is generally fucked. When I used to work with homeless youth, those not suffering from at least mild depression always raised red flags for me because, given all they faced, if they weren't somewhat depressed, or having a behavior disorder then likely they were masking something much deeper.
Which leads me to the second category of people potentially dealing with psychological duress: Some people don't have what is necessarily "damage" but are, as you say, non-neurotypical (though that term makes me cringe). Folks who might be diagnosed with bi-polar, clinical depression, schizophrenia, schizo-affective (aka when old enough they'll be diagnosed with schizophrenia), and those on the autism spectrum might fall in to this category.
How we deal with them would likely depend on which of the two categories (or both) and what combination of behavior you are dealing with. Generally speaking, everything Animalevolent says is good advice. Some more particular things that I have gleaned and which may or may not be useful given your situation:
1. Don't pretend to be a mental health professional if you aren't. Diagnosing others isn't helpful, and if done in any public manner, could be the equivalent of badjacketing them. If there is particular behavior that needs to be addressed, address that, but don't throw around terms that are often diagnoses as pejoratives.
2. Stay flexible - the DSM IV and the diagnoses in it are merely ways of generalizing about individuals psychology, based on observed patterns. Schizophrenia looks different in different people, and it is not a particular set of symptoms, but a correlation of a certain number of attributes that *might* be schizophrenia. The same is true of most other diagnoses. That means not everyone reacts the same to the same coping/treatment/wellness plans. Someone might seemingly present with something, and that is not what it is. Also, they might have a particular diagnosis and not react the same way as others with the same diagnosis.
3. Be real - I used to know a guy who was severely schizophrenic. I don't mean that as a pejorative sentence - he was kind, creative, and an amazing poet and artist, as well as someone with the potential for the most intense empathy and sensitivity that I have encountered. He also saw geometric patterns that spoke to him, and at times viewed himself as a messianic. When I first met him, I would go along with whatever he said so as not to "upset" him. Over time, I realized that he actually found it more helpful when I would just tell him, "I don't think I follow you." A common acquaintance once indulged one of his hallucinations of voices, at which point he looked at her with wide and confused eyes and asked, "you hear them too?" He was not, in that interaction, looking for validation of his experience, but a touchstone as to what others are experiencing. Falsely claiming common experience is doing no one any good. On the other hand, the young mand who had a "ghost" that helped him through his daily routines and gave him positive reinforcement about himself so he could (hopefully) get through his day didn't need people actively telling him that the old man ghost wasn't real, seeing as how this was one of his most positive relationships.
If someone you interact with is in need of further care than you can provide, to help them get that help would hinge on what is happening right then. If possible, getting them to a quiet place with less people (not necessarily alone, though most people in sever psychological distress aren't a threat to anyone, it is a good idea to have someone to have your back) is going to help. Try to roll with what is going on, but also try to assess how much consensus reality you share; knowing this will be crucial to connecting effectively. Use easily digested words and sentences, without condescending. Express whatever concerns you have for their safety, and see if you can come up with a plan with them for what to do. Depending on where they are at, this might not be to get "help" so much as it is to cope or stabilize for a bit. Since you can't and presumably don't want to force them to get any help they don't want, the best option is to help them in the moment and address it later.
There is a lot more I could say about this, but I don't know how much is relevant or necessary.