That difference doesn't exist when you talk to people about their perceptions of mental illness and violent behaviour. They think that people with a mental illness are likely to be violent, even though that's very unlikely. They think that a violent event will have been perpetrated by a person with a mental illness even though again it's not likely.
That's their problem then, not the problem of the parent author. If you think video games are real life and shoot someone because Call of Duty made you do it, you have a mental illness. No where did anyone here make the claim that everyone with a mental illness would do that though.
I drink tea because I have a cold. Not everyone who has a cold drinks tea, though. I'm guessing that mental illnesses might be a sensitive subject for you for some reason, but being overly politically correct isn't the answer to solving the social stigma. You can't deny that people who have trouble separating fiction from reality have a mental illness.