What do the Silent Hill f difficulty modes mean?

Let's start with explaining Silent Hill f Puzzle difficulty. There are three options that are basically easy, default, and hard, confusingly named like this:

Easy - Story
Default - Hard
Hard - Lost in the Fog

You can get an example of this to some extent in the Silent Hill f locker codes puzzle, which features the clues for the harder options alongside the default difficulty. So while the 'hard' option features clues like converting 'bus' via an alphanumeric code to 805, the harder options has clues like 4151+3√6. The puzzle where you have to find the correct Silent Hill f ema, on the other hand, changes between finding three animals on Story/Easy and deciphering a slightly more complicated poem on Default/Hard.

Hard is basically the default option, in that puzzles will take a little bit of working out but aren't actually 'hard' hard, if you see what I mean.

The Silent Hill f Action difficulty options is harder to explain, as it doesn't actually explain any of what changes with your choices at the start. There are also only two options to begin with, as the third isn't unlocked until you complete the game once. However, once you start tutorials appear occasionally to explain combat and also clarify how your difficulty choice affects combat. From that, this is how Silent Hill f Action difficulty changes things:

Story difficulty lets you recover both Health and Sanity at a Hokora (shrine) without using Faith (gained from trading in collectables). In other difficulties you'll need to use Faith and consumables to recover Health and Sanity.

Max Sanity will not decrease when attacked while using Focus in Story mode to deal extra damage. In other difficulties, max Sanity will be lowered if you're attacked while using Focus.

Being grabbed in Story mode will not decrease your max Sanity, but it will in other difficulties.

So Story mode mainly protects your Sanity, which can be depleted by events, monsters, and actions in the world - making it harder to lose and easier to replenish. In Hard you'll lose it more easily and find it harder to restore, and when Sanity is depleted you'll start to take Health damage instead.

Once you've finished the game you'll then unlock a further harder Action difficulty called Lost in the Fog, like the Puzzle option, which is tougher still.