I have no idea it can vary system to system. The problem with windows 7 and bf4 i believe is that windows 7 does not automatically unpark the cpu cores for best performance in game like windows 8 does. There are many tutorials on how to unpark cores on windows 7 but you dont have to do that on 8. If i were you i would get 8.1 its the newest and i have read the 8.1 users will get a free upgrade to windows 10 when it comes out.
I find 8.1 to be better for everything performance wise and yes i have heard that bf4 runs better on 8.1 as well. The only problem is most people (myself included) dont like the windows 8 layout, the lack of a start button the stupid touch screen friendly tiles on the home page etc. but if you can get past the aesthetics i think (and this is just my opinion) that 8.1 is the better OS.
I have no idea it can vary system to system. The problem with windows 7 and bf4 i believe is that windows 7 does not automatically unpark the cpu cores for best performance in game like windows 8 does. There are many tutorials on how to unpark cores on windows 7 but you dont have to do that on 8. If i were you i would get 8.1 its the newest and i have read the 8.1 users will get a free upgrade to windows 10 when it comes out.