White rouge is usually aluminum oxide. It's roughly 50% harder than chromium oxide, and will be a better polisher on steels with a little more wear-resistance (think of steels like 440C, 154CM, VG-10 and beyond, up to something like D2 or ZDP-189 in wear-resistance). Chromium oxide, a.k.a., green compound, isn't quite as effective on more wear-resistant steels like those, and does better with low-wear steels like 1095, CV, 420HC, 440A, etc. White rouge is often a bit larger in grit size than chromium oxide (0.5-1 micron is common for green); the Ryobi white rouge I use is rated at 2-5 microns. White rouge will be a lot more aggressive in polishing, working very fast with the steels I mentioned. It works very well on a hard-backed denim/linen strop, in particular. It'll leave the edge much more crisp at a high polish, with more bite, than will green compound on those moderately wear-resistant steels, which tend to get over-burnished and rounded off at the edge, if stropped too much with green compound.
That said, on steels with much vanadium carbide content, at ~4% or higher like S30V or M390, something like diamond or CBN will handle those better for polishing to shaving-sharp edges, as white rouge, green and other compounds won't be hard enough to polish or refine vanadium carbides.