That’s funny, Mary Anne, because I would have said the opposite, that “How are you” sounds more casual. Here in Massachusetts (America), I often hear that used very casually, like by the cashier at the check out line.
blueygh2, to answer your question, to my ear it means the exact same thing, and people always answer it as if it is (although they’d be more honest if it was asked sincerely, than if casually). In the statement “How are you?,” there is an implied “doing?” at the end. I don’t even know what it would mean to ask someone “How are you?” without that implied doing. Are there other implied meanings in other countries?
“How are you feeling?” is also implied. If someone isn’t feeling well emotionally, they aren’t likely to answer, “Great.” They might say “Okay,” or “Pretty good” in a flat tone, if they don’t want to get into the details with a stranger.