The common mineral gypsum softens soil and allows lawns to become revitalized. Sugar, however, has not been proven to have a positive effect to soil or lawns.
Gypsum is not a fertilizer, so there is little chance it of it harming lawns. Gypsum comes from all over the world and is mined for many uses; everything from tofu to chalk may contain some of the mineral.
Sugar, on the other hand, might alter the pH levels of the soil. This can cause damage to the natural cycle of growth by stunting it, as the natural absorption of nutrients is inhibited.