Weight matters when it comes to sound insulation, but only incidentally. The true measure of soundproofing is density. The more dense materials are, the more material sound waves need to pass through in order to move beyond the insulation, and the more easily more sound waves will deteriorate, deadening the noise. The more dense the material, the greater its weight. So while weight itself does not affect sound, greater weight is a side effect and often a measurement of better sound protection.
The relationship between weight and density leads to simple, effective solutions for increasing the soundproofing qualities of rooms. An additional layer of foamboard can give the walls greater thickness and more weight, but may also keep the walls from letting as much sound through. When choosing insulation materials, the heavier, denser materials are able to reduce sound more than lighter, potentially less expensive materials.
Completeness is a sound insulation factor connected to how well the insulation is installed. The type and density of the insulation is important, but its quality also depends on full installation without any cracks, gaps or flaws. A gap between insulation boards, for example, renders any density benefit futile, as the sound waves easily escape between the boards. Completeness is an overarching factor you should consider no matter what type of insulation you use.
If you cannot increase the weight and density of your insulation because of cost, materials at hand or deadlines, consider using flexibility and isolation factors to improve soundproofing instead. Flexibility refers to how rigid the materials are. In some cases, rigid materials can actually reverberate sound more loudly than flexible materials, even if they are heavier. Isolation refers to the difference in materials between the soundproofed room and nearby spaces. Switching isolation materials, avoiding flanking transmission through hidden links, like support beams, and keeping materials flexible are all alternative methods of increasing soundproofing.