how much difference is practical terms is there btw f2.8 (and below) and f4? (more specifically, those super exp 2.8 lenses vs the also quite exp f4 ones. namely the 2 versions of 70-200mm L)

Practical terms meaning :

1. Since its brighter, HOW MUCH darker or faster are u able to achieve good AF using the 2.8 vs 4.0? Would the fact taht u have a focus assist (or even better external speedlite)

[im asssuming just just coz it allow twice the light to get in, DOES NOT MEAN u can AF twice as fast, or twice as dark rite? im sure the lower DOF of the 2.8 also affects AF... so its not a linear relationship]

Anyone have any practical values? (eg. In street lamp light, 2.8 can acheve AF 50% faster, or can af 80% of the time vs f4s 50%)

2. 2.8 allows DOUBLE the light of the 4.0 right? so basically this would mean shutter speed can be TWICE as fast as if u were to use a 4.0? (correct if im wrong)

3. Bokeh wise, are both equally good?

4. distortion wise at full wide, both as good?

Thanks!!