To understand this, you need to understand a little about the little sod that causes white spot. It has quite a complicated life cycle. The bit we're most familiar with is the white spots on our fish. These are small nodules each of which house a "trophont". This eats away at the fishes cells and body fluids, it is constantly wriggling about and irritates the fish no end.
Once it is mature, it erupts from the cyst and attaches itself to something fixed, (rock, plant etc.), where it builds a strong skin around itself. This stage is called a "tomont". This is the principle replication phase. Inside it's protective sheath, it divides and divides and divides. Eventually, the skin ruptures and many of the new, free swimming stage tomites start swimming around trying to find a fish. There can be 1000's from a single cyst.
Note, the cyst on the fish, and the encapsulated tomont are pretty much indestructible. It is only the free swimming stage that is killed off by medications. The reason the heater is often screwed up a little when treating Ich is not to kill the parasites, but to speed up the life cycle and get them to the free swimming stage faster.
Now you know that, lets consider some scenarios.
First, it is possible there was no white spot in the tank at all, and the new fish introduced it. Possible, but I suggest unlikely.
Second, after a treatment, there are always a few parasites at the trophont stage that enter a dormant phase. They stop growing, stop feeding, stop irritating - they're there - but are not doing anything. Once your stressed Neon start releasing, (albeit indirectly), tomites, a chemical signal triggers the previously dormant trophont to resume activity. Spot breaks out in the hosts where previously they were in check. These dormant trophonts normally congregate in the gills, or at the base of fins, it is for that reason that the original fish often seem to be intensly irritated as those trophonts in the gills are in particulaly sensitive places.
Third, the very few tomonts in the dormant stage release free swimmers at intervals, but their number is sufficiently low for the immune response of the normal fish to reject them. Along comes your stressed Neon with trophonts popping out like corks, a few days later, there are now tens of thousands of the free swimmers in the water, even the tougher fish in the system are overwhelmed by the onslaught.
There are probably other scenarios as well, but consider, you do not need to introduce new fish to a tank to trigger an outbreak. An established tank of many years with no change of occupants can suddenly have an outbreak of white spot if the tank gets chilled, or some other stress factor is introduced. The little sods are there looking for an opportunity...
Imagine having your body covered with, (to scale), egg sized parasites wriggling about on or just under the surface of your skin. I'm starting to itch just thinking about it.