I know that the moon has lunar regolith and earth has earth soil, but what is the difference between them?
The single biggest difference is the lack of chemical weathering in lunar soils which are subject to physical weathering almost exclusively. If you exclude biological processes, terrestrial rocks undergo significant weathering from water and atmosphere, which the moon lacks.
For example, both earth and moon contain feldspar-rich rocks, however, clays, the result of chemically altered feldspars, are not found on the moon. Neither are oxidized minerals, as the moon has no oxygen-rich atmosphere to speak of.