I’ve seen a number of discussions of the difference between mögen and gefallen but I haven’t come across anything attempting to differentiate passen and gefallen which seems curious to me as unlike mögen and gefallen, these two share the use of the dative case to refer to the person doing the “liking” or who is “suited” or “pleased by” something (.e.g, using mir for “me” in these examples):
Es gefällt mir besser
Das passt mir besser
One of the meanings that pons.eu gives for passen is angenehm sein – “to be pleasant or pleasing.” There is a noun das Gefallen – “pleasure” – that pons indicates can be used with haben or finden to mean “to get pleasure from doing something.”
Both gefallen and passen seem a bit less direct than mögen, perhaps because of the use of the dative case, and thus it seems like they might be used in cases where you want to soften a negative statement. As an English speaker my intuitions may be way off, but it feels as though instead of saying “I don’t like it” fairly directly with mögen, you could say “It doesn’t suit me” with passen or gefallen. I will need to try this hypothesis out on a few native speakers to see if it gets some “likes” (oder Gefällt mir à la Facebook)!