ffmpeg: How to convert quicktime (iPhone) video to mp4 format?
I'm new to ffmpeg, but if you want the codecs to remain the same and the data wrapped in mp4 instead of mov, I believe the following will work:

 ffmpeg -i video.mov -c copy video.mp4


The command you posted may have worked just fine, but your computer is having trouble playing the video data. This is almost always a codec issue.

You can compare the data in the mov input with the mp4 output using ffprobe. Compare the output of the commands ffprobe video.mov and ffprobe video.mp4. If the above command works as expected, then the data should read the same under the codecs. Only the container related data should be different.