My main reason is convenience. My laptop did a good job, but I had to boot it up, select the right program, max the screen etc etc. This was in XP Pro, so I'm assuming something like XP Media Center with a remote control, or perhaps a dedicated homebrew solution would be better. Oh, and my laptop gets pretty hot after a few hours.
The other thing is the hardware in some of the boxes (I have Tvix and Buffalo) is dedicated to playing video. From what I've seen of my computers (several laptops and desktops) it just doesn't quite hack it regardless of the codec it seems. This ranges from the odd freeze to lip sync. Now fair enough this can happen on the media devices too, but from what I've seen the computers do it more often. Especially when the mandatory Spyware and virus killers do something to freeze the film for a split second (not just for full sweeps but also for just updating the definitions). This is not to mention X other applications that can kick in at any time. If there is a program that could dedicate everything to the playing of video and completely stop all other tasks then great, but I don't know of it.
The other reason is price. I mean my laptop was pretty good when I got it, but cost 6 or 7 times as much as my Tvix and about 10 times more than the Buffalo.
Finally from my experimenting with these devices I found the media players generally better as long as they are playing their supported codecs (eg. DivX on a Tivx is pretty solid). Ok, so a new codec comes out (eg. H264) and your media player is toast for those files whereas your computer is fine as long as it has the processing power, but then again just convert these files to the supported format (Xillisoft has some pretty good software, as well as WinAVIVideoConvertor which is quite good). One example I had was a film that completely glitched for about a second then totally lost the lip sync until you stopped and started again manually. This happened on a few different computers running on different video software. All did the same thing pretty much. On the Tvix there was the slightest of lines on the screen for like a 100th of a second with no lip sync problems at all.