The gravity associated with any object is not related to the origins of the object, so even if the moon was originally part of Earth (something which is still unclear), there is no reason it would have the same gravity as the earth. Instead, the attractive force of gravity between two objects depends on the masses of the objects and the distance between them, and is given by the equation F = G*(m₁*m₂)/r², where m1m₁ and m₁ are the masses of the objects, r is the distance between them, and G is a constant called the gravitational constant. (G=6.674*10^-11 m₃/(kg-s²). Note that this is different from the better-known acceleration due to gravity g=9.81 m/s².) So we can see that objects with more mass will exert more force from gravity, but that force will decrease as the separation distance increases. By comparing the mass and size of the earth and the moon, we can see why the gravity acts more strongly on something on the surface of the earth than on the moon. _{The mass of the earth is ~100x that of the moon, but the radius is only ~4x that of the moon. The higher mass offsets the increased distance, and the force of gravity is greater on a person on Earth than on the moon (by ~6x).}

So the really short answer is that gravity is higher on earth than on the moon because the earth has a lot more mass and is only a little bit bigger across.