I personally really like the fat bar for pressing movements (military, bench, incline bench, floor press, close grip bench mostly). I notice that it is easier on my shoulders and the larger surface seems to allow for a greater for transfer to the bar.
As Granite posted earlier, I’m not fond of it for pulling exercises. As he mentioned, it limits the amount of weight you can use. YES it will strengthen the grip. But personally I prefer to do grip work by itself and use a regular bar that allows me to lift more weight.
I feel that occasional use of the fat bar for pulling movement can be useful. For example I sometimes use it as potentiation:
When I ramp on a big pulling exercise, I sometimes do the lighter sets with the fat bar. And then switch to the smaller bar. This both potentiate the involved muscles and also makes me feel more confident when i switch to the smaller bar. Pierre Roy (canadian weightlifting coach) used to have the women start lifting on the men’s bar (28mm versus 25mm for the women’s bar) then switch to the women’s bar when the weight got heavier.
I also like the fat bar for barbell hip thrusts and Zercher squats.