The heaviest objects don't necessarily have the largest volumes. Volume is the amount of space an object occupies, and "heavy" refers to its weight, which is proportional to mass. If two objects have the same density, then the heavier object will hava a proportionately larger volume. Density is mass divided by volume, and has a constant value for a given material.
If two objects have different densities, then volume is not an absolute indicator of mass. Volume and mass are not directly proportional in this case, so any assumptions are arbitrary. A golf ball and an inflated balloon is an example of a pair of objects with different densities in which the heavier object has less volume. A ton of bricks and a feather is an example of a pair of objects with different densities in which the heavier object has more volume.
We can use mass and weight interchangeably if the force of gravity is constant, as it is in the same location. Mass doesn't vary with location, but weight does because it is mass x acceleration due to gravity. In other words weight is the force an object exerts when gravity acts on its mass.