How to convert a .mov file to mp4 file?
Use FFMPEG

It's open source, free and its libraries are what Handbrake (and others) use to encode MPEG file formats.  If you are going to have to use their software anyway, why not get it from the source?

You can download it from here:  http://ffmpeg.org/download.html

Once you have it installed, just issue the command in Terminal:

ffmpeg -i input.mov output.mp4


Where input.mov and output.mp4 are the actual file names and paths of the files you want to convert from and to respectively.

I used to use Handbrake and VLC in the past to do this sort of thing, but ever since I discovered FFMPEG command line utility, I have no need for either apps any longer.