They are both very quiet. I don't think I've ever heard either one of them in real usage.
Getting the shield to see/use anything external is frustrating. I haven't tried to access external media via Kodi. If this is a real issue, the chromebox is probably better. Just share it over the network and avoid plugging it into the shield.
The shield is truly plug and play. I've used (many) PC's, Rpi2+Openelec, chromebox with plex and Kodi, and now the shield. Nothing comes close to the simplicity of the shield and I'm a huge fan. You can have Plex and/or Kodi running in literally minutes. Plus, it can legally stream anything except for (maybe) Amazon Prime. As an added bonus, you can play silly stuff like bombsquad with your friends when you have a party. It's hilarious. Hint: be the player with the real game controller and tell your fiends that their smartphones are just as good (they aren't)!