They're very close. The ISO base media file format was directly based off of QuickTime container format.
If you look at a .mov file and a .mp4 file in a ISO bmff viewer, you'll generally see the only difference is the ftyp box is different ("qt " for .mov, "isom" for .mp4). Indeed, if you ask ffmpeg to make a .mov file and .mp4 file of the same content, literally the only difference is the contents of the "ftyp" box, every other byte is identical.
If you look at a .mov file and a .mp4 file in a ISO bmff viewer, you'll generally see the only difference is the ftyp box is different ("qt " for .mov, "isom" for .mp4). Indeed, if you ask ffmpeg to make a .mov file and .mp4 file of the same content, literally the only difference is the contents of the "ftyp" box, every other byte is identical.