There are reasons to use both 30 and 60fps. 30fps gives more of a classic look, and you can depict fast motion better (albeit less clearly, as it will blur). Most movies/shows are shot at 24-30fps.
60fps gives the fake/soap-opera effect (some people more sensitive to it than others) on slower scenes, but if you want clear, fast, motion it looks better when shooting action. If you've seen the new Hobbit trilogy, that was shot in 48 FPS (twice that of a 'normal' film) and looks like absolute **** on still scenes (props super obvious, everything looks fake, etc.), but great for the action scenes, so it's a trade off. That effect is even more noticeable at 60fps.
With video, each frame is basically a photograph. How many of those per second is what fps is and is directly correlated with motion blur.
Note 60fps is only available in 720p and 1080p, 4K is limited to 30fps. The slow motion 120/240 FPS are in the side menu along with Panorama, Photo Sphere, etc.