What are the differences between the Gamecube and the Wii version of Twilight Princess? Are they just the same game, only with another control scheme (Controller for Gamecube, Wiimote for Wii), or are there any significant differences?
The main difference is that the whole game is flipped, horizontally. In the Game Cube version, Link is left handed (as he was in all previous games). In the Wii version, he is right handed. In order to avoid re-doing all of the graphics in the game, they just flipped the entire game - Lake Hylia is in the southeast corner instead of the southwest corner, if there is a path to the left in the Game Cube version, it is to the right on the Wii, etc.
I recall that Nintendo stated that this had something to do with players using the Wiimote in their right hand, and the interaction with the game was odd if Link was left handed, but I'm not positive on that.
According to Wikipedia, they were released at the same time, mainly just with a different control scheme. However, since most people are right-handed, and Link was left-handed in the GC version, they decided to flip the entire world, changing West to the East and East to the West. So, just watch out for that when using a Walkthrough or trying to play on multiple versions.
To add to other answers, there is one badass thing that you can do on Gamecube version, but not on Wii version:
Try loading your spin attack while on Epona's back. This makes Link raise his arm and sword in a very nice, stylish effect, with some sparkle going along the sword. When charging toward enemies with the B button pushed, it felt like some LOTR vibes that I wish they hadn't removed from Wii version.