Typically, the biggest advantage is just that it looks far better. 60 fps is much, much smoother and more fluid than 30 fps. I hear some people can't tell the difference, but that seems odd. I suppose it's possible though that if you've been playing/watching at 30 fps all your life, you may not immediately notice the difference between 30 fps and 60 fps, but personally I can see differences in frame rate as low as 10 fps without having to look at a frame counter. Here's a test to see if you can tell the difference:
http://boallen.com/fps-compare.html
With fast paced first person shooters, ideally you want a higher frame rate. Let's say you're running BF4 at 60 fps. Now let's pretend that an enemy comes into your character's view from the left side of the screen. From the time an enemy moves into view of your camera, the time it takes for you to see him appear from the left side of the screen would take a minimum of 16.7 milliseconds (60 fps means 60 frames/second, hence 1000 milliseconds/60 frames = 16.7 ms/frame). This is because technically, the enemy comes into view of your camera 16.7ms earlier, but your graphics card, if it's running at 60 fps, needs 16.7ms to actually draw the image.
Now let's take 30 fps. At 30 fps in the same scenario with an enemy coming on screen, it would take a minimum of 33.3ms to spot an enemy from the time he comes on screen. Sure, we're dealing in milliseconds here and so that might not seem like a huge amount, but in reality (assuming your perception is good enough in the first place) you notice this hugely. Essentially, for competitive games a good frame rate is advantageous because it allows you to react quicker. Of course, it also looks and feels better too.