60fps can be converted to 30fps, but 30fps can never be converted to true 60fps (just every frame will be shown twice).
The main difference between these two framerates are aesthetics and feel.
Most people are trained to associate certain framerates to certain media. For example, movie theaters are showing their films in 24 frames per second, and many people believe that with this low framerate comes a big part of the “movie look and feel”.
50 or 60 fps are highly used in television, some in interlaced formats (50i, 60i), many people associate with those framerates a cheaper, television-soap-kind of look, although technically it is a more advanced technology.
What makes the difference in these viewer experiences is not only the amount of frames shown, but also the shutter speed these frames are captured with, which is most of the time directly related to the framerate. Usually, you expose one frame half as long as it stays on screen, e.g. with a framerate of 30fps, your shutterspeed is 1/60th of a second. With 24fps, your shutterspeed is ideally 1/48th of a second and with 60fps you probably go with 1/120th of a second.
This differences in exposure lengths highly affect the achieved motion blur. Things in motion recorded at a long shutter like 1/48th will have a longer motion blur trail, and look much more blurry, than a video recorded with a shutter of 1/120th.
Sport events, where fast moving objects are common, are often filmed in a higher framerate, because the resolution of the motion is much higher with a higher framerate and faster shutter speed.
Many people refer to the super-sharp, not so motion-blurry look as the “videolook”, opposed to the movie look.
The question to what framerate you want to use, or what shutterspeed you should use, can be answered by thinking about in what area your footage will be used in most likely, and what kind of things you record.
For static shots, you don’t really need 60fps, as the amount of motion does not require such a high framerate.
It all comes down to personal taste, aesthetics, feels, and what people are used to look at.