I cant be exactly sure but what I think they meant by "soft" is that they live in soft water. They have very exacting needs for long term culture. I also think that it could be that because most of these fish in the hobby are wild caught they have a harder time getting used to new conditions. If you look them up it is recommended to have water at like 5 to 6 pH and soft.
Cardinal tetras which look a lot like true neons are less finicky about conditions IME.
I had neons in hard water at one time. Only half of them lived for a good while, but they never flourished and never had great color. I'm pretty certain it was the water being too hard. With softer water, lower pH, they are supposed to be very hardy fish from what I've read and heard.
Neons are a softwater species but considering that they are one of the most popular species in the aquarium trade they will live in just about any water that is in good condition. The water where I live is normally on the alkaline side and could charitably be called 'liquid rock' but neons are sold here by several LFS and kept by people who don't necessarily know about softening water. I've kept them myself although now I do use an acid buffer and black water extract to soften the water and lower the pH. More information on neons can be found at http://www.fishprofiles.com/files/profiles/577.htm. I've found that cardinal tetras are more sensitive to water conditions than neons are.