The other part is that whereas the flu knocks you out, mental illness can make you harder to predict, which is potentially more dangerous. Some of that stigma is probably built in by evolution.
Stigma is almost entirely a cultural thing: I know of someone who's first manifestation of mania (bipolar disorder, type 1) was in a religious setting... so when she said she heard God talking to her, they were reportedly thrilled. It didn't come out that it was a problem until she was on the plane home and was blessing the passengers (edit: details slightly obscured/changed).