Actually I have watched this same debate involving neons and cardinals go on in the 1960's! It was one I remember and so I've watched it with interest over the last couple years here on TFF and within some searches even.
My current personal observations are as follows. Despite a couple of rounds among experienced keepers here, where we (me included) have speculated that the "sensitivity" of neons might just be down to cases of trace ammonia and nitrite and/or slightly high nitrate in tanks that are not really as fully cycled as we like to get them.. we've always continued to get the observation of people losing neons, on a small percentage basis, even with tanks that we judge to have good water parameters.. but are new. Here's the thing: Its just an "observation," not a thing that has ever connected up with a "cause."
One of the major problems with it is that its just a fairly small percentage thing. All along, there continue to be observations like JoshuaA describes, where neons are introduced to a new tank but they do just fine. You get enough of those and you begin to feel its just down to the usual water things when along comes a case where no one can find fault with the water but the neons just croak. Even in the 1960's I remember some experienced fishkeepers saying to just introduce them to a much more mature tank and you'll almost never experience a death. There's just some other factor about neons that's unrelated to our usual water things and we don't know what it is, I feel. I feel our collected "observation" (as hobbyists in the large sense) is correct to some degree, but I feel no one (at least in my experience) has really had anything to point us in the direction of some different factor that the neons are sensitive to.
If you look at the case of Dwarf Gouramis, the hobby went through a period of mystery about unexplained deaths but then finally a micro-culprit was found and the syndrome was explained to a large degree. It could be that something like this will turn up with respect to neons but for now my feeling is that it doesn't quite "feel" like this.. partly because the reports of total success in more mature tanks seem to go so solidly up to 100%. I know to younger hobbyists, this it bound to have a touchy-feely aspect when they read it but it you'd watched it for a long time you might be surprised to end up with a feeling similar to mine.
Now whenever the neon thing is discussed, I feel it also needs to be noted that losing neons to transport continues to be an observation widely reported. One thought from a scientific standpoint that plays in to this is that if you look at the evolutionary coping strategies of different fish species I believe one can observe a difference in strategies where fish have larger body mass and have developed greater "survival machinery" within the individual fish, VERSUS, shoals of fish with very small body mass who have developed survival strategies with a greater emphasis on shoaling and great numbers. Think for instance of the "light reflecting" trick that neons and other irridescent scaled fish have - its been figured out that they can cause light to reflect and interact with surface water reflections in a way that literally fools larger bottom and mid-dweller predators to think the prey is in a slightly different position - the predator misses the mark, going after the reflection.
Anyway, I do feel that many very small mass fish do not match larger mass fish in their capability to survive -as an individual- but that they do in fact survive just as well when their gene pool is looked at -as a group.- Of course, that gets us no closer to clues about what environmental detail might be the major factor in the higher percentage demise of neons under transport or new tank conditions, if indeed that non-scientific would really hold up if we ever had actual science done on it, which is unlikely of course.
Ah well, just my drop of water in the sea.. ~~waterdrop~~