In simple terms, balances measure mass, while scales measure weight. So if you bring a balance and a scale to the moon, the balance will accurately give you the mass of, say, a moon rock, whereas the scale will be affected by the gravity.
A balance determines mass by balancing an unknown mass against a known mass. In modern weighing instruments, balances usually use a force restoration mechanism that creates a force to balance the force exerted by the unknown mass.
A scale displays weight by measuring a deflection; the springs are deformed by the load, and the force needed to deform the springs is measured, and converted into weight.
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