They are closer than they look
MP4 was derived from the QuickTime file format that MOV uses. Structurally they are near-siblings, and in the common case an iPhone MOV and an MP4 contain exactly the same H.264 or HEVC video and AAC audio.
That means converting MOV to MP4 is frequently just a container change: the streams are copied across untouched, no quality is lost, and it takes seconds rather than minutes.
Where they genuinely differ
MOV can carry things MP4 cannot handle as gracefully - ProRes and other high-bitrate editing codecs, multiple timecode tracks, and some Apple-specific metadata. A ProRes MOV from a professional camera is a different animal entirely and will be enormous.
MP4 has broader support for streaming-oriented features and is the format upload pipelines are built around.
HEVC (H.265) complicates things in both containers: it halves file size but is not universally decodable, and an HEVC MOV from a recent iPhone is a common source of "this file won't play" complaints on Windows.
The practical rule
Keep the MOV as your master if it came from your camera or editor - it is the closest thing to the original.
Convert to MP4 whenever the file leaves your own machine: uploads, sharing, clients, collaborators.
If a converted file still refuses to play somewhere, the culprit is usually HEVC rather than the container. Re-encoding to H.264 fixes it at the cost of a slightly larger file.
The short version
Archive the MOV, publish the MP4. The conversion usually costs you nothing in quality because the video inside is already the same.