Pete is considering the f4 70-200 because he has a back problem, i.e. it may be 'better' for his needs - not because he thinks it's a better lens. If he could easily carry the 2.8, I've no doubt he would continue to do so.
It's unarguably better to have available a max. aperture of f2.8 than one of f4, especially when the f2.8 version is one that is frequently, and happily, shot wide open.
The sole advantage the f4 has is 'convenience' (which to be fair, IS important if it meant that otherwise you simply would not have a 70-200 in your bag).