From system Menu 4.0 and up, SDHC cards are supported. As for most homebrew apps, SDHC support is added. I myself use a 4 GB SDHC and everything works fine.
Only problem I've had, was the ability to copy wii saves to my SDHC. But using Wii Game Manager GX solves that problem, it's a homebrew app with SDHC support. Does just what you need. Copy, restore wii saves to your system.
You'll be fine with an SDHC. If you can, keep both. I have a spare 2 GB SD with my Brawl hack files on it. More memory is better.
Overall, there's very few things that don't work with SDHC, but they're very minor and have alternatives. Mostly everything, works with SDHC. Especially important things.
Hope that helps!