As far as I remember, Resident Evil Archives has the Rocket Launcher Ammo glitch fixed (but that was only present on the US and JPN versions of GC), and some small changes in some voice clips that weren't brought over to the HD versions. About Resident Evil Zero Archives, it uses a different font for certain texts, and maybe has another change here and there.
As Leo Alves pointed out, Wii Versions support Progressive Scan (480P), and besides the Wiimote+Nunchuk and the Classic Controller, I've read that Archives also support de GC controller. The GC versions also support 480P (for REmake I think you needed to hold B when launching the game), or can it be forced through Nintendont (only US REmake, not PAL), making the first point moot, and if you load from USB, disc-swapping is also a non-issue.
Oh, and by the way, don't play Archives with the system set up at 16:9. For some reason the way game handles a system on 16:9 is to put black borders, reducing the vertical resolution at which the game is rendered making it look worse. So you're forced to change your system to 4:3 if you want to play it, or rather, force it to 4:3 in your loader of choice. GC version doesn't do this, it displays at 640x480 and looks at it should.
Overall nothing too important as far as changes to the games themselves. So play whatever version you can find, both are fine (Archives 16:9 issue notwithstanding), only real plus is that Wii has other controller options, but the game was designed for a GC controller... In fact, the GC controller is quite important, as the Classic-Style Resident Evil games released on GC had a very, very nifty feature: Control Type C.
Instead of using Up to move forward, you use the R trigger, making changing directions while moving smoother, and also you avoid unintentional moving while rotating, but not only that, depending on how deep you're pressing the trigger, you will run or walk, so there's no need for a Run button. It's the most comfortable way to play for me, and is great that I can also play RE2, RE3 and Code Veronica X with that setup.