I hear the newer ones do not play GCN games. How can one distinguish these consoles? (Date of manufacture? Console color? Bundled games? GameCube controller port panel? Something else?) For instance, what questions would I need to ask an eBay or Craigslist seller if the seller didn't have GCN games to try?
The original Wii supports GameCube games on disc, and has GameCube controller ports hidden under a plastic door.
Wii Family Edition
There is an "updated" Wii that goes by "Wii Family Edition" on the retail box. It is identical in many ways to the old style Wii, down to the form factor. However, it does not support GameCube discs or controllers.
The best way to tell them apart is the model number - the Family Edition is model number RVL-101, and the "classic" Wii is model number RVL-001. This number is printed on a label on the bottom of the unit when it is in the vertical orientation. If it's lying horizontally, this label is on the right side of the unit when looking at the front.
You can confirm this information on Nintendo's support website. Click to view the Wii manuals, and you'll see two sets, one for the RVL-001, and one for the RVL-101. The 001 manual mentions the GameCube ports, while the 101 does not.
In addition to the model number change, there are a couple of other telltale signs:
Wii Mini/Wii U
Since introducing the Family edition, Nintendo has also released the Wii Mini in some markets, which does not support GameCube games at all. It is also missing network support, and has a top-loading disc drive rather than the front-loading drive of the original Wii design. It is essentially a bare-bones, stripped down version of the original Wii.
The new iteration of Nintendo gaming hardware is called the Wii U. Despite similar names, and cross-compatibility with certain Wii peripherals, the new Wii U does not support GameCube discs either.
There's a way to put Gamecube ports back on the Wii Family Edition. You can switch the parts around so the Wii Family Edition will play Gamecube games and the original Wii can't. This is the only way to play Gamecube games on the new Wii. You should do this at your own risk.
Xzibit is also disappointed by the news. There was not a definitive resource that I could find, but it seems a tell tale aspect of the new design is that it is "designed to sit horizontally". Each mention of the new design also includes a statement similar to this one:
It also seems consistent that the new consoles all include a Wii logo that is upright in the horizontal position as seen here:
as compared to:
You might gain some confidence by checking for serial numbers mentioned in this article. Since those are older consoles that predate the shift to dropping GameCube support.
However, the best idea is to just get a picture of the GameCube controller ports. If it has ports, it has GameCube support.