Aside from being smaller, the Moon has a distinctly different bulk composition from Earth--its mantle is proportionately much larger, and its core is proportionately much smaller.
Suppose that Earth had the same primordial composition as the Moon, just scaled up in mass 81 times to match the real Earth's mass. The resulting planet would surely be much more like our real Earth than like our real Moon, if only because it would be able to retain whatever atmosphere outgases from the early mantle, rather than becoming a dry, airless rock.
But would anything necessarily be significantly different from Earth? (Specifically, excluding the sort of accidental differences that you would get from re-running an alternate Earth with identical chemical composition, like incidentally different continent shapes and such.)