I'm imagning that these plug in through USB ports since they are fairly recent purchases (as opposed to nine years old). I'm also assuming that at some point, you have plugged them into the same port to make sure that there's nothing wrong with the particular port you're using.
Really, only three possibilities come to mind, although I'm open to ideas from others on the forum:
1.) The non working dvd player is not getting power for some reason. Being plugged in to a wall, or any other power source for that matter does not automatically equal getting power.
2.) The non working dvd player is rather seriously broken (it's probably not handshaking over the usb connection if there's no "bah doomp" sound right when you plug it in, assuming that the other one does make the sound -- can't think of any easy fix for this).
3.) The USB cable going from the computer to the non-working one is bad for whatever reason. This doesn't happen all that often, but it sure is a cheap and easy fix when it does.
Again, it's entirely possible that someone else has some better answer, but this is what comes to my mind.