The Best Way to Convert an MOV file to an MP4 File is with Davinci Resolve 

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Are you finding the quality of your videos that were converted from .mov to .mp4 to bad? I used both VLC Player and Handbrake to convert a mov file to an mp4 file and both of the files that were created had a really poor quality when I brought them into Camtasia or when I watched them in the built-in Windows Movie Player.
I was really surprised by this because I wasn't seeing the same issues when I was experimenting with Davinci Resolve. Davinci has both a free and paid version, but the free version will do pretty much everything you need to do with your videos including outputting or rendering them to mp4 format.
I'll be honest, I'm not a fan of using Davinci Resolve to edit my videos because I find the controls non-user friendly, but as far as converting a file from MOV to MP4 it does an excellent job.