The law of conservation of mass, which states that in a closed system, mass is neither created nor destroyed, it can only change form. This means that in a chemical reaction that takes place in a closed system, the mass of the reactants equals the mass of the products.

The law of conservation of mass/matter states that mass/matter cannot be created or destroyed in chemical or physical changes.

Law of Conservation of mass(atomic mass).

As mass can be considered relative to energy, therefore Law of Conservation is also correct but Law of conservation of mass is is much more accurate because here mass is a much more accurate term that is required here.

Here, since, we are balancing molecules, then we require atomic or molecular mass.

it is the same as the law of conservation of mass

Law of mass conservation in chemistry: in a chemical reaction the mass of reactants is equal to the mass of products.
Law of energy conservation: in a closed system the energy remain constant.