Why is the weight of an object different on earth and the moon even though the objects mass is the same in both places?

Mass is an intrinsic property of matter, it does not vary.

Weight however is variable, it is a measure of the pull of gravity on a mass.

As the Moon's gravity is less than that of the Earth, a mass will weigh less on the Moon than it does on the Earth.

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Because the mass of the moon is different from the earth's.

Also their radii is different.

(Note : weight = gravity x mass

and gravity = m1 x m2/(r x r)