How do you generate difficulty?

[Quertus]

on the other hand, in most sandboxes the party will still find level-appropriate encounters. Some DM will probably have the sandbox say "and there is a dungeon here", but only fill it with appropriateely leveled denizens when needed. or they will find a way to telegraph which adventures are appropriate and which are not.

Most? That may be true, but does not match my personal style of sandboxes, nor my experiences with the sandboxes of others (where "ancient red dragon" was on the random encounter table, and was rolled with 1st level PCs).

Telegraph? I'll agree that, in the modern age, doing so is considered good form; personally, I'm not a fan, and prefer "discoverable".

It is also worth noting that a sandbox is generally filled with a few plot hooks, and the players are expected to latch onto one of them.

Well, there's a lot of nuiance missing here. Let me try to massage this a bit.

One of my sandboxes might contain, say, a dozen plots. Of which, by the end of session 1, the players might be aware of about 3. But the plots aren't designed with the explicit concept of "plot hooks" - it's up to the players to decide whether they want to get involved, and what they want to do about a given plot. Rumors of goblins in the East evolving, and seeking further improvements? The party is welcome to go (try to) investigate, wipe them out, aid them, con them into buying new false upgrades, steal the source of their power for themselves… or just ignore the situation entirely.

All in all, I would say that my sandboxes, at least, are filled with many plots, and the players are expected to latch onto one or more (preferably more) and/or develop one or more (preferably more) of their own.

What makes it a sandbox is that a) all content is optional, and b) there is no "prescribed method" of engaging any given content.

if they don't, and just keep doing random stuff, the dm is likely to get bored and quit, or to start railroading them.

… I guess it depends on what you describe as "random stuff". If the PCs decide that they want to set up a taco stand in "goblin evolution central", then that's them both engaging in one of the existing plots, and creating their own plot, not them doing "random stuff".

If, OTOH, they have a table of responses, and choose to <roll> ignore <roll> ignore <roll> ignore <roll> play a game with the fourth NPC <roll> ignore <roll> ignore <roll> slaughter the seventh NPC? Yeah, I'd probably get bored/quit.

and that, unless the dm is highly disfunctional, even a railroad will give plenty of freedom to the players; not much in what is their goal, but in how to accomplish it.

This is clearly a difference in definition of terms. To me, the act of "railroading" involves exactly that dysfunction you speak of - the GM negating/perverting the logical consequences of an action in order to force/preserve their script.

And a sandbox is often not as well established as it's supposed to be.

What do you mean by this sentence?

so, the distinction between a sandbox and a railroad is often more blurred that what is often assumed.

A lot of people use railroad & sandbox as a binary distinction, but I think they're more usefully thought of as ends of a continuum. Very few, if any, campaigns are purely one or the other.

I am glad that the Playground developed the "sandboxy" nomenclature to describe the somewhat continuum/spectrum nature of reality in this regard.

I probably have a false trichotomy in my head of "railroad", "sandbox", and "normal" games.

In a normal game, there is a plot, and plot hooks, which may have been sunken into the PCs in session 0. If the players somehow miss the plot, it represents a fail state.

In a sandbox game, there is no single plot, and "plot hooks" are incidental. If the players fail to engage any of the content, and do not create a plot of their own, it represents a fail state.

In a railroad, the GM has a defined script, of not just "what" but "how" (and "when"). Any deviation from that script represents a fail state (which will likely invoke the response of "railroading").

So, that's the false trichotomy that lives in my head, and what I'm usually thinking when I hear or use those words.