You can't compare them
The "Disarm" action is not exactly a variant rule, but a combat option. Combat options are described in the DMG as "alternative ways to handle combat". That is to say, "Disarm" action mechanics is mostly for a DM, that's why it is not described in the PHB.
In its turn, PHB has the improvised action description:
Improvising an action
Your character can do things not covered by the actions in this chapter, such as breaking down doors, intimidating enemies, sensing weaknesses in magical defenses, or calling for a parley with a foe. The only limits to the actions you can attempt are your imagination and your character's ability scores. See the descriptions of the ability scores in chapter 7 for inspiration as you improvise.
When you describe an action not detailed elsewhere in
the rules, the DM tells you whether that action is possible
and what kind of roll you need to make, if any, to determine
success or failure.
Characters can attempt to make any actions they want. A player can say "I am trying to knock the lantern out of his hand". In this case the DM can use the existing mechanics from the DMG:
A creature can use a weapon attack to knock a weapon or another item from a target's grasp.
Alternatively, DM can use a different roll, or just say what happens, without any rolls.
On the other hand, the Disarming Attack maneuver is described in the PHB as a strict class mechanics. It is a distinctive feature of the Battle Master archetype, which the Fighter player should be aware of. In their turn, the player should reference to this exact maneuver, in orter to be sure how it will work mechanically.
Disarming Attack. When you hit a creature with a
weapon attack, you can expend one superiority die to
attempt to disarm the target, forcing it to drop one item
of your choice that it's holding
That's it - when you hit you can say "I expend a superiority die" and disarm the opponent. You say it as a player to the DM, not as your character.
The main mechanical difference
Of course, before the game starts, the DM can say "I use an optional rule for disarming. Any character can disarm an opponent with an attack roll". In this case the Disarming Attack maneuver still has a major advantage.
A player announce the Disarming Attack maneuver after the attack roll, and the common Disarm action before the attack roll.
When you announce the Disarm action you forgo any other action and still can miss, effectively wasting the action. In contrast, when the Battle Master's attack hits, they can announce the Disarming Attack maneuver. Keep in mind that the Fighter class also has the Extra attack feature.