Brain health sounds less serious to me for some reason, like you have physical and you have mental health. Brain health is like the way you would explain the meaning of mental health to a 5 year old.
I think when it comes to mental health itself people have understood it much better, there's less of a stigma. However when people talk about someone they saw in the newspaper going on a shooting spree or something and say he's "mental" and people who use "mental" as an insult or "mentally unstable". It's like people who say "are you retarded?" or "are you dumb?" or "are you autistic?" as an insult. It doesn't mean retarded/ dumb/ autistic people are thought of in a bad way, it's just the use of the term in the different contexts. People know what autism and mental retardation are, and they know they're serious and they understand that, and that is not affected by the usage of these terms in those contexts.