As I understand it, it wasn't actually stainless steel vs. something else (stainless steel hadn't yet been invented, anyway). It's more an issue of the shape of the blade in cross-section: the "old-style" knife was the same thickness top to bottom (and then the edge of this was sharpened, so in profile it would be trapezoidal); the "new" one (called סכין מלוטש in the literature) gradually tapers towards the cutting edge.
R' Shneur Zalman of Liadi discusses the matter in a letter to the rabbis of Vilna (an excerpt of it is also published in his responsa appended to his Shulchan Aruch, no. 7).
He reports, first of all, that R' Chaim Volozhiner quoted the Vilna Gaon as saying that there's actually no halachic problem with using polished knives; the ban against them was due solely to the early concerns about the legitimacy of Chassidus. So once it was recognized that Chassidus is not (G-d forbid) a heterodox movement, then that reason fell away.
RSZ goes on to explain that there is in fact a halachic advantage to using such a knife. There is a chumra, based on the opinions of some Rishonim, that it must not have any detectable nicks at all (the opposing view is that small ones are fine, as long as a fingernail doesn't catch in them). With old-style knives, getting the cutting edge this smooth would usually come at the expense of its being less sharp, while with tapered knives it is easier to make the knife both sharp and perfectly smooth. So as this chumra became more widely adopted, the new knife naturally came along with it.