Wasing with sand is still done today. Soap is generally though of as "chemical"removal of dirt and oils, while sand is "physical" removal of dirt and oils.
Because sand is fine, coarse, and usually not too absorbent, it will rinse right off. There are many common products today that work the same way. Lava Soap, is a good example.
Sand is also often used in cleaning cookware. For example when camping you can use sand to clean a cast iron fry pan. You can not use soap, and salt may be too damaging to the environment (in a large concentration).
Washing with sand, (and not water or soap) is a good way to "clean up" when water is scarce. It will remove dirt and oils, help with sweat, and generally leave you cleaner then using nothing at all.
That said, your reading a religious document so there is certainly a religious reason behind using only water. Unfortunately, I don't know enough to really answer the why.
Only that washing with sand and water, or only sand, are both viable and in use today, though perhaps soap and water (or only "soap" if we include disinfectants) is more common.