No there is no difference between a black wii and a red wii. However there is a difference between a standard wii and a wii family edition. The difference being: the wii family edition is designed to sit horizontally ( the standard wii can do this too ), comes bundled with a game; like New Super Mario Bros, wii; and more importantly does not have GameCube controller or memory card slots, and therefore is not compatible with GameCube games.
So, there are two different wiis, standard model number RVL-001, and wii family edition model number RVL-101.
The main difference is that the family edition is not backwards compatible with the GameCube.