Bingo. We already see this with HDMI and HDCP where you can get a diminished experience, or none at all, because the playback device and the TV can't get their ducks in a row.
Amazon will serve me HD video on my smart TV using its Amazon app, but not on my computer using the same TV as a second monitor. And lest you think the problem is the computer, I can get it on my PC just fine using a regular computer monitor...it's a bummer to think about this sort of thing with simple audio as well.
Huh. I often use my non-smart TV as a second monitor, but I'd never tried playing an Amazon video on it. Sure enough, the same video that plays in Firefox on a VGA monitor gives me a "we're experiencing a problem playing this video." message if I move the browser to the TV.