DND_Reborn
The High Aldwin
Citation?A human or human shaped creature with makeup (so, most races in 5e) will absolutely far more damage with a staff used two handed rather than one handed.

Of course you can deal more damage with two hands than one (d8 versatile vs. d6), but I would not say it is "far more damage". Much of the damage depends on where the fulcrum of strike originates from as well as where and how the blow lands. A straight thrust with the butt-end of a staff can cause severe damage, breaking ribs, etc.
Also, many one-handed strikes have one end of the staff braced under the arm and against the back/side:
Although "one-handed", by having one end of the staff braced by the back or side, the fulcrum is towards the far end, allowing the tip to travel a longer arc (just as quickly in many cases) and hit with greater force than a staff held with two hands which are closer together if holding the staff towards the center. This effectively move the fulcrum towards the middle, making the arc much shorter and thus hits with much less force.
As I said:
Anyway, swinging a quarterstaff one-handed is a perfectly valid technique as well. As @loverdrive mentions, the force at the end is greater due to the acceleration the tip has passing through a longer arc.
The full video is here:
Someone who, especially, takes the Polearm Mastery feat, should be able to make one-handed strikes. The butt-end (or other end) bonus action strike would have to be executed more when holding the staff towards the center, further decreasing the force, thus doing only a d4 damage.