Container and codec are different questions
MOV and MP4 describe containers that can hold video, audio, and other streams. A container is not itself a guarantee about how the picture was encoded. A MOV may use a codec that the receiving application accepts, or one it cannot decode; an MP4 can face the same issue if its streams are unsupported.
If the goal is compatibility, identify the application's requirements for video codec, audio codec, resolution, and frame rate. Renaming a .mov file to .mp4 changes none of those streams. A real conversion may repackage them or re-encode them, depending on the workflow, so test playback in the intended destination.