What makes neutral spirits "neutral" isn't what it's made out of, but how it's made.
Neutral spirits (Grain Neutral Spirits or GNS if made from grain -- but NS can be made from anything fermentable) are distilled at nearly 100% ABV and are, well, neutral. No flavor at all. The better vodkas don't quite reach this goal, which is where their subtle flavors come from, but commercial NS is usually right up there in the 195 proof area. As a source of flavorless alcohol (not to mention cheap, of course) GNS is handy for blending with more flavorful spirits to produce a "lighter" beverage. If the blended spirit is to be whiskey, the neutral spirits MUST be GNS.
"Sugar spirit", essentially a kind of rum if made with reconstituted cane sugar juice (i.e., sugar and water) can, like any other fermented-base product, be distilled to whatever degree of purity the maker wishes. Typically the moonshiner is not concerned with the extra expense of highly pure distillation, certainly not so in the case of a distiller using sugar to stretch his mash (or to BE the "mash" as is true in many cases). Sugar spirit that is NOT distilled all the way to tastelessness usually tastes pretty awful. Think cachaça. Bad cachaça. The sugar used can just as easily (and more cheaply) be beet sugar, which isn't even suitable to be called rum.
Bottom line: If you use any non-grain-based spirits in the process you can't legally sell your product as whiskey, straight or blended. For products less concerned with legal approval, cane sugar is a common alternative among home-made liquor folks and some artisan distillers, but not that much with real 'shiners. After all, if what you want is lots of alcohol and you want it fast, why bother with converting grain at all? I would guess that the "thin" part of a thin mash more often refers to a small amount of cornmeal thrown in to add just a touch of actual flavor (you'd need a LOT of cornmeal to actually mask the sugar flavors) or so you can "honestly" call it corn liquor.