Excuse my ignorance with all of this...
I understand your confusion. And this is something I grapple with daily and beat myself up over. There IS an emotional component to PTSD. That's obvious and hard to ignore and that's what people most often recognize. But there is a component for me and for some others that is less emotional in nature and more physical response. This isn't me trying to make myself 'feel better' about this. It's actually the thing that my therapist keeps drilling into my head when I have flashbacks : this is not something I have any control over. This is my brain being hijacked.
The ADA defines a person with a disability as a person who has a physical or mental impairment that substantially limits one or more major life activity.
I actually wrote more extensively about what my SD does for me in a different thread. He isn't there just to "make me feel good and be cute" That kind of IS what an Emotional Support Animal does. The difference is the training. An ESA does NOT HAVE ANY TRAINING. They are just there to be a companion. They also do not have ANY PUBLIC ACCESS. Believe me when I say that even learning to very simply stand in a position and lean against me on cue (and more often without my prompting but because he knows that I am anxious) take a ton of training.
1. grounding work (providing me with what people like to call deep pressure therapy. It's hard to stay dissociated with a 90 lb dog on your lap or leaning against you)
2. Blocking (i.e. he will stand between me and other people.) It helps keeps me from having a panic attack standing in line, say, at the grocery store. Before I would wind up running out and abandoning my shopping because I couldn't deal with people being behind me. He pretty much does this automatically when I start getting anxious. He will stand behind me and lean in on my legs.
3. He also performs something called "cover me" which is when he will be a presence and behave as though he is watching my back while I can't watch it. He is NOT trained to attack. He is there as a deterrent. I KNOW this but there person I fear might be behind me does NOT know this.
We are working on 'who's there',(for reality checking) and searching the house (though he does this quite well already) since I have serious problems entering my home and needing to check every nook and cranny to ensure it is safe.
Finally, we are working on him recognizing and breaking flashbacks/really bad dissociative spells. He does this by recognizing the change that comes along with a flashback and then disrupting it by vigorously licking my hands/face etc to get my attention. That tactile feedback breaks through that.
It's been a game changer for me.
There are other things that a PTSD dog can do but these are specifics that MY dog does for me.
This is a useful link to help explain more.
http://www.iaadp.org/psd_tasks.html
This is actually a great list of possible tasks that SD's can do for someone. Many of them don't make sense for me. For instance, I don't NEED a dog to remind me to take pills. I don't NEED a dog to fetch water so I don't train those tasks but for someone else, that would make perfect sense.
I'm not sure if any of that helps you understand what the difference is or not. My service dog mitigates my disability. Yes, it's taken quite a while to use that word with any ... comfort. I do not look disabled. Some part of my brain however IS.