In addition to Pokemon Battle Revolution as noted in ChrisF's answer, Animal Crossing: City Folk allows you to connect to it's DS counterpart (Animal Crossing: Wild World) and import characters. Also, My Pokemon Ranch (as stated by Michael Madsen) is a WiiWare game that functions similarly to Pokemon Box for the Gamecube - it acts as a very large storage box for your DS Pokemon games.
If you connect the DS version of Geometry Wars: Galaxies to the Wii version, you can unlock the final galaxy. Castlevania: Order of Ecclesia (DS) and Castlevania Judgment (Wii) connect to unlock additional content as well.