Basically, you can use the exact same metal and don't need folding for good enough metal. Same with the soft hard method...
Folding is just to even out the impurities in the steel. With pure enough steel, like modern steel, folding is pointless. I think that technically it does improve the durability by changing the shape of the crystalline structure, but that isn't maintained through the hardening process, as heating it up changes the structure back to "default". I'm not sure about that part, though, and I'd have to look it up to confirm/debunk.
The soft/hard method is a bit different, though. Spring steel has a uniform hardening which can flex quite a lot, but if it flexes too much, it snaps. Spring steel also requires a relatively more pure steel to not be too unreliable. I think that's because the tempering process is quite precise, and impurity can change that too much, but I'm not sure.
Differential hardening on the other hand doesn't flex (well, it does flex, but not significantly), but deforms and bends with too much force. However, that can be bent back, so the sword isn't completely destroyed (unlike with LotR shows, you can't reforge a sword without completely redoing the entire process).
You also get a harder edge, which means it will retain a sharp edge longer, but it's more brittle.
You get these advantages and drawbacks regardless of steel purity, since the rely on the hardening and tempering process.