NeoGamma allows you to use cheats.
SoftChip allows you to choose which IOS to use, it's very helpfull for Rock Band for example. But this is only an advantage over NeoGamma, cioscorp includes a cIOS37.
NeoGamma and SoftChip allow you to play import games that require patching to the language of the game. (mostly japanese games)
NeoGamma and SoftChip allow you to use both wii and disc region video mode. Disc region video mode has 100% compatiblity and basically allows you to play every compatible game from whatever region it is. Wii region allows you to play games in color if you tv does not support the other standard, but only has a 50% compatilibty)
NeoGamma and SofChip can easily use latest cIOS updates like the rev8 for dual layer games.
My toy(look at my sig):
With the next relase, everything that SofChip does plus it allows you to patch video modes. This usefull if your tv does not support the other standard and you play import games. But it only achieves a ~50 success rate which results in ~75% of import games that can be played with correct colors.
Cioscorp allows you to play games from the disc channel without using a loader and with it you don't need to IOS Version patch games for NeoGamma or SoftChip. The downsides are:
1. It can't yet be installed of "unsoftmoddable" Wiis
2. It's heaviliy messing with the system's firmware and even slightly changes how retail games are read from disc.