A lot of year or older cameras do not support SDHC cards. But if they aren't more than 4 or 5 years old, then there is probably a firmware upgrade for your camera. Which, once upgraded, will allow SDHC to work fine.
As for a Wii. A lot of games won't recognize an SDHC card. Then again, some games won't recognize any type over 2 GBs. But it will work for most other things, like Channels, as long has you are running System Menu 4.0 or higher.
Also, if you are trying to "mod" your Wii, it definitely won't work.
My personal experience with an SDHC card...
Had to upgrade the firmware on both my camera and my digital picture frame in order for it to work.
On the other hand, of the two printers I have, one Epson and one HP, that both have multiple types of memory card reader slots. Neither one will recognize an SDHC card, and there isn't a firmware update to fix this issue.
So printing images straight from the card won't work (which is the main purpose of the one printer). The work around is to hook the camera up via the USB cable, that came with it, to the computer, and then transfer the images off the card that way.
Bottom line, if you are tech savvy, I think you'll know enough about whether or not an SDHC will be ok for you.
If you aren't tech savvy, I would suggest staying away from an SDHC card... unless every thing you own, that you might use it in, is brand new.