There are five moves for which weight is a factor: Low Kick, Grass Knot, Heat Crash, Heavy Slam, and Sky Drop.
For Low Kick and Grass Knot, only the target's weight is taken into consideration, and only for the purpose of calculating how many predefined jump points that weight exceeds. In some cases, being heavier (with Heavy Metal) will cause a Pokemon to cross an additional jump point, and therefore take more damage from those moves. Being lighter (with Light Metal, a held Float Stone, or use of the move Autotomize) can take it down across a jump point so it takes less damage from those moves, but not always: for example, Metagross is so heavy that even Light Metal only reduces its weight from 550kg to 275, still enough to exceed the last jump point and take maximum damage from Low Kick and Grass Knot.
Heat Crash and Heavy Slam care about the relative difference between both Pokemon's weights, and the jump points for these moves are based on thresholds like "weighs twice as much as the opponent," "weighs three times as much," and so on. When it comes to these moves, being heavier means you often deal more damage as the user of the move, or take less damage as the recipient, but there are times when the weight disparity is already large enough not to make a difference (for example, pretty much any Pokemon that uses Heavy Slam or Heat Crash against a Ribombee will hit for maximum power regardless of weight-modifying abilities or items). A Celesteela using Heavy Slam will get the greatest power threshold from the move (never mind a possible type resistance) against any Pokemon with Ho-oh's weight or less; Celesteela has been known to use Autotomize primarily for a speed boost, and even with the weight reduction effect, there are only 10 Pokemon that avoid the 120-power range as a result, leaving Raikou as the heaviest one still in that bracket. Copperajah with Heavy Metal can become even heavier than Celesteela and reaches full power against targets that weigh as much as Terrakion or Gigalith (and not coincidentally, it has access to both Heat Crash and Heavy Slam).
For Sky Drop, there is only a single, hardcoded threshold of 200kg (the same as the final jump point for Low Kick and Grass Knot, and the lightest weight that avoids taking Celesteela's Heavy Slam for maximum power). If a Pokemon weighs this much or more, opponents will be unable to pick them up with Sky Drop (no matter how heavy they are), and will simply waste their turn. Among Heavy Metal users, Lairon and Bronzong are both in the right weight range where Heavy Metal pushes them over the boundary to gain this immunity; for totem Pokemon, Araquanid and Kommo-o (both of whom are otherwise weak to Flying) also get it if you use the totem version instead of the regular version. Aggron is already heavy enough to have Sky Drop immunity without needing further benefit from its ability, and while Cufant would be in the same bracket as Lairon and Bronzong to obtain this benefit, Sky Drop is not a selectable move in any game where Cufant exists and this interaction is not a factor for it.