You said that non-organic potatoes would not grow. I don't think that I have ever tried to grow store bought potatoes as I've been growing Yucon Gold that I bought from a seed exchange a few years ago. However, I have seen ordanary store bought potatoes sprout when I've forgot about them in the pantry.
Well they might but you would have to keep them out for a while and get them sprouting. I remember reading an article somewhere that recommended not to use non organic. It actually recommended against any store bought due to the risk of introducing disease as I alluded to but I had to find out for myself.
After about half a day to ponder this issue, I've come up with this: Genetically Modified Organism (GMO) potatoes (if GMO potatoes exist) are probably the ones to avoid. I know many GMOs are built not to propagate. Thus forcing the farmer to go back and buy from the seed manufacturer every year. So making the assumption that all organic potatoes are non-GMO, this may explain the problem your running into with non-organics. However, I think the more accurate dividing line is GMO/non-GMO.
Now, moving onto disease, I don't see any real reason why there would be a difference, but I never really thought about it. My gut tells me that the disease issues likely to occur are mostly rot. Which would be seen before you even plant. In the end this is an issue that is be well suited to experimentation, anyone ever try growing store bought, non-organic, non-GMO potatoes?
From my readings and viewing, GMO is not permitted in organically labeled food. Whether or not this is properly regulated may be debatable but I have heard this from multiple sources.
In regard to store bought potatoes versus certified seed, below are a couple references discussing the subject. As you can see from my pictures, my plants look very healthy but the store bought plants haven't been harvested yet. The red pontiac were purchased at a nursery so maybe I'll do a follow up post in a month and let you know how my store bought turned out.
http://homeguides.sfgate.com/tips-planting-potatoes-store-68919.html
http://www.gardeningchannel.com/growing-potatoes-from-store-bought-potatoes/