Why do we use an object's mass, rather than its weight, to indicate the amount of matter it contains?

Because mass doesn't changed based on location and scientists rely on a property that is constant no matter where the object may be.

Mass is the amount of matter an object contains (density), where as weight is the measurement of gravity's affect on that object ... For example a person weight more on Earth than on the Moon because Earth has higher mass, and thus higher gravitational pull. While mass is the amount of matter some body is made of.