If I take a kilogram of lead and a kilogram of feathers, obviously they weigh is the same, but if I take a certain volume of feathers and the same volume of lead, then the lead will be heavier.
The volume and density affects the weight, pressure depends on what you compare it to.

Barometric pressure of a mercury instrument is compared against the vacuum that the falling metal creates.
In an open environment, warm air has less density than cold air, therefore it has less molecules per cubic area, less molecules means less weight, less weight pressing against the moving surface of the barometric instrument indicates less pressure.