That extra stop adds a great deal of size and weight.
For walk-around shooting the difference is immediately noticeable, and it is more than just size and weight alone; it is the balance on the camera along with the sense of needing to support the weight of the 70-200/2.8; this feeling does not occur with the 70-200/4, nor does one feel that the lens mount is at risk of (invisible to the eye) damage with a pendulous heavy lever arm.
The weight different is not quite double.
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