You can rent movies on a Mac Mini, just not through Front Row (you have to go through iTunes with a keyboard or mouse) so it doesn’t make for a good TV companion. The great thing about the Apple TV is that you can do everything with the remote, so it will really blend in with your setup. Also, the graphics card in a Mac Mini doesn’t support a high enough resolution output to get the most out of your HD television. The Apple TV will allow you to plug a HDMI cable right into the back and get going.
Looking at your situation, I would say go with an Apple TV. It’s a little cheaper too, so you can use the leftover cash to buy some more really great sofa cushions and a bean bag. Have fun.