Mental illnesses are you yourself
Normal illnesses you can externalize. You can have an injury in your foot, a herniated disc, an ulcer, a cold sore, or a urinary infection, and they're all "out there." Those problems aren't you, they're things you have, that you suffer from, but you can think of them as being located outside yourself. They can affect you and maybe make you sad or frustrate you, but you can talk about all of them in the third person.
Mental illnesses aren't like that. Mental illnesses are you yourself. They change you: your perception, your decisions, your fears, your intellect, your plans, and your life. They're like a diabolical spell that alters your being from within, a pair of glasses that color everything you see and everything you do. That's why it's so hard to notice when they attack you. You don't see them coming, and you can often be settled into them without even knowing it.
Once you start to realize how they operate, they create a permanent doubt: is this tension me, or is it the anxiety? Is this lack of enthusiasm normal, or is it the depression? There's no clear separation between the illness and you, because it blends with you, it dissolves into your mind like milk in coffee. In the end you start to learn how the symptoms come and go, and sometimes you can attribute to the illness something you used to think was your own fault.
This is reassuring, because until then you lived thinking you were useless for not being happy, for not taking things more calmly, or for having to spend the afternoon in bed out of sheer apathy. You wondered: how does everyone else manage to be okay with so little effort? If the illness is you, then it seems the blame is yours when your behavior is abnormal. The problem is that the people around you make that same deduction.
Others, too, confuse you with your illness, so they interpret your actions as genuinely coming from your own will, and maybe as defining of who you are. And I'm not talking about people you barely know; they can be members of your own family. If it's hard for you to distinguish between the real you and the illness, it's obviously going to be far harder for them, because they don't have access to your thoughts. So anxious people are told to relax, depressed people are told to cheer up, and in general the advice you get is the kind that goes: "you just have to see things differently." Sometimes you even hear someone say that suicidal people are selfish.
All of this, from the point of view of someone with a mental illness, is a monumental injustice. The feeling of being misunderstood by everyone is the plot of the best dramatic movies, and it produces, all on its own, even more problems. On top of the difficulties of living with it, of the debilitating internal struggle, your diabolical spell makes you look bad and people think you're like that because you want to be.
And even though you realize that, deep down, it's natural for it to be this way, this difficulty in separating the sufferer from the illness is unique to mental illnesses, and it's the root of much of the stigma and of the ignorance of those who don't experience them.