First: I think this is a great post. Beeing interested in other consoles is something many gamers are not able to.
Okay let´s start:
Many people already mentioned Twilight Princess and in short: It just feels better swinging the Wiimote than pressing a button ever felt. Of course there is no 1:1 motion in the game but imagine you could actually swing your Wiimote in the next Zelda game. It just feels great.
Mario Galaxy is another good example: In this game you control a little cursor with your remote and if you scroll the cursor over several things in the game world you can interact with it. You can grap enemies, make rockets stand still or click on little blue stars to make Mario fly through the galaxy. In many games the Wiimote feels pretty tacked on and this is true. But good developed games use the Wiimote to intense the experience, games like Mario. In good developed games you don´t even notice there is something "revolutionary" cuz it just feels intuitive. While you had to use the control stick to use your bow in previous Zelda games you can just aim the screen now, which works much faster and more accurate.
The Wiimote also gets you better posibilities to change your playstile: you don´t have to hold your hands in front of you - actually I let one hand lie down on one side of my body. This is very comfortable and shaking the Wiimote doesn´t need much of a movement.
Using the Wiimote to play racing games is much better than usign a control stick. And every Review about Call of Duty and Medal of Honor on Wii I read said the games controls are much better than with a control stick (still they are not as good as a mouse ---> gamepad < Wiimote < mouse).
Playing a game while using gestures is pretty fun and once you played Zelda on Wii you don´t want to go back anymore.
And of course: the Wiimote is something new and you can change the way you play (Nunchuk, Classic Controller, no attachement) which is something you couldn´t do before and holding a controller in one hand is pretty cool too.
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