I feel like I'm walking into the lion's den coming into this debate so late.
Anyway, this is a tough one. On the one hand, the Gamecube was probably the last console I had while at a young enough age that I was susceptible to truly having my mind blown by things I hadn't seen before. I miss that childlike sense of awe. That's not to say that I'm such a miserable, cynical bastard now that I'm no longer impressed or excited by anything - on the contrary, some of my favourite games ever have been in the last few years. But you know that feeling you got when you were younger, when you see something for the first time, like a Super Mario 64 or an Ocarina of Time? I really hope someone understands what I'm talking about. It was also at a time when I would much more commonly have friends over to play multiplayer games. The fact is, there was a point in my life when I would play Smash Bros. Melee every day with my mates from school. I have more memories of those kind of experiences with the Gamecube.
On the other hand, the Wii has been one of the forerunners of this 'retro revival' wave that has been gradually sweeping this generation of gaming. This generation, 2D platformers have come back in a big way, and so have numerous other genres that were thought to have died one or two generations ago. Old franchises are back and they all have that familiar feel, but with a crisp, fresh modern sheen over them. In a time where shooters and big budget action titles rule the day, the Wii is a safe haven for a good old platform challenge, or an adventure, or a clever puzzle. And when I look at my collection of Wii games, I can point to a decent number and say that probably wouldn't exist on another platform.
That's my stance. I'm not going to pick which is better; ultimately it would be a pretty arbitrary decision.