I think the distinction without a difference is a tool being deterministic or not. Fundamentally, its nature doesn't matter, if in actual practice it outperforms everything else.
And for that matter, going back to the band saw analogy, a measure of a quality of a great band saw is, in fact, that the blade won’t snap in half in the middle of a cut. If a band saw manufacturer produces a band saw with a really low binomial p-value (meaning it is less deterministic/more stochastic) that is a pretty lousy band saw, and good carpenters will know to stay away from that brand of band saws.
To me this paints a picture of a distinction that does indeed have a difference. A pretty important difference for that matter.