FXAA is inferior in quality but much, much less demanding than MSAA. If your computer can handle it you should go for MSAA, it's clear that the quality should be better once you understand how each of the antialiasing methods work. Of course, I'd rather have 20 fps more in bf4 than have that 10% increase in quality, there is a price/performance ratio which FXAA wins.
on a monitor with HUGE pixel density sure. but on a completely normal display playing games without any kind of AA - meh. I'd rather have 30 fps avarage
Firstly I must say that 'quality' in this case IS subjective.
MSAA does have its benefits though I believe in certain implementations where shader AA is an issue and where TRAA isn't applied FXAA is a lot more beneficial.
Even when removing geometry aliasing FXAA can provide a smoother experience.
Thankfully bf3 and 4 allow for the best of both worlds. That said, so many people avoid FXAA and I find that hard to understand given my reasons in the OP. Compared to straight up MSAA, FXAA isn't just the performance route.