Due to a debate I saw earlier, I thought it's for the best if we compared GameCube and 3DS graphics side by side. Just so we see how good/bad these newfangled games look. I'll be comparing similar games, including entries in the same series whenever possible. Note that the bigger 3:4 screenshots will be the GameCube ones, whereas the smaller 3:5 ones will be for the 3DS.

First let's do some Mario franchise series, starting with Luigi's Mansion.

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I choose these two images because they both portray the series-specific character E. Gadd. Note how, other than the different lighting, the two Gadds have similar features, down to the swirly glasses and freckle locations.

Next, let's look at some 3D platformers! Super Mario Sunshine vs. 3D Land.

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Each of these images have multiple characters in a arger setting. Note that Sunshine has a more "plasticy" look, for the better or for the worse.

Now for some Mario Kart! Double Dash vs. 7.

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It's two versions of Rainbow Road (couldn't find a decent one of the new 7 Road). Both look pretty nice to me.

Next is time for a couple of Mario Party games. I chose MP4 out of the four Mario Party games for the GameCube, simply because it came first.

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I personally think the 3DS game looks more bright and colorful.

Now for some Paper Mario! Thousand Year Door vs. Sticker Star:

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TYD looks more impressive, but I think SS has a better art direction.

One more Mario game: Tennis!

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My point goes to the 3DS game for the better backgrounds.

Now for a semi-Mario game: Smash Bros!

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I think the 3DS Peach simply looks better than the Melee one.

Now for some non-Mario series. I was going to do Animal Crossing first, but seeing as the GameCube version is an N64 port, that wouldn't be fair. SO let's skip to the Pokemon games!

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This one is weird. The GameCube had WAY better backgrounds, but the 3DS has WAY better Pokemon.

Well, onto 3rd party games! Let's start with Resident Evil.

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DAMN I forgot how good Revelations looked!

Next is Super Monkey Ball: The original vs. 3D!

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...I honestly don't know enough about these games to compare their graphics.

Next is Harvest Moon: Wonderful Life vs. New Beginning

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Other than coloring, no big difference. The 3DS game is a lot prettier though.

Last, we have Sonic: Heroes vs. Lost World

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Sort of like Mario, the 3DS game looks less like plastic, for better and for worse.

I'll be adding more games and screenshots later, but I think the 3DS shows more impressive tech/graphics than the GameCube.

So GCN wins then?

The 3ds looks better in most cases imo but it might be due to superior art direction rather than polygon count... Interesting for sure

The smash bros one really shows the difference

But that picture of Princess Peach on Gamecube is just ewww... I don't remember her looking like that at all

Interesting. Based on these screenshots:

Luigi's Mansion 2 > Luigi's Mansion 1

Super Mario Sunshine > Super Mario 3D Land

Mario Kart: Double Dash > Mario Kart 7

Mario Party 3DS > Mario Party 4

Paper Mario 3DS > Paper Mario GC

Mario Tennis GC > Mario Tennis 3DS

Revelations > Resi 4

I'm undecided on the Pokemons. That screenshot of Colesseum or XD is hideous; but X/Y aren't really lookers either. I'm undecided about the Smash games too. Objectively I think the 3DS version looks better, but the black lines around all the characters is really offputting.

Two things to note: Quality of screens are not equal among all of these pictures. Also, screens are inherently going to look worse for the 3DS than the Gamecube for some obvious reasons.
Although I would say that the screen for RE: 4 seems to be low quality.

.....God damn it, ugh why does these kind of conversation keep popping up...it's just....so pointless. Don't handhelds and home consoles use completely different architectures. Why do you think it the Wii version of Donkey Kong Country Return look so different from the 3DS version. IF the 3DS share similar tech and was more powerful under the hood compare to the Wii, porting the game wouldn't have taking so long.

I still maintain that it's a tie, but damn, seeing that particular screenshot of Sunshine makes me want a 3DS port even more...

Edit: That being said, Sunshine was the last Mario game that took itself seriously and attempted to move the series forward, not sure if comparing it to a nostalgia haven is completely rational, but it's negligible.

Basically it seems to be 3DS does better closeups, but when games have a wide open area, they seem to go to the GC.

But if 3DS can do GC stuff... all I want it F-Zero 3D!!!! NOW PLEASE!

Resi 4 shits on Revelations on polygon count and art style, for me.

I think that the screen resolution is in part to blame for that.