Weight
In modern scientific usage, weight and mass are fundamentally different quantities: mass is an intrinsic property of matter, whereas weight is a "force" that results from the action of gravity on matter: it measures how strongly the force of gravity pulls on that matter. However, in most practical everyday situations the word "weight" is used when, strictly, "mass" is meant. For example, most people would say that an object "weighs one kilogram", even though the kilogram is a unit of mass.
Kibble balance
In addition to measuring "UI", the laboratory must also measure "v" and "g" using experimental methods that do not depend on the definition of mass. The overall precision of "m" depends on the precision of the measurements of "U", "I", "v" and "g". Since there are already methods of measuring "v" and "g" to very high precision, the uncertainty of the mass measurement is dominated by the measurement of "UI", which is the value measured by the Kibble balance.