As a writer, I prefer to give what you call the mix of the two. But really what it is is the illusion of free roaming. And so while this is true for stuff I publish with various companies, it also holds true for GMs. There's a few ways to pull this off.
The most common way is that the GM lays out three (or whatever) plot hooks for the PCs. Maybe they overhear about a lost city that's starting to become uncovered in the nearby desert; a fig merchant has a problem with his shipments being hijacked as they head over the mountains out of the protection area of the city-state; and while all this is going on, the city-state's house guards are waging a battle against drug dealers who are rumored to live in the sewers, but nobody can ever find a sign of them when they go and look.
So now the PCs have three hooks. Each hook has a different locale, and each locale is likely inhabited by different stuff. The lost city is a great place to use undead or evil outsiders, and maybe a few giant scorpions; the fig merchant's problem is going to resolve in the mountains, and that could be anything but lets go with some sort of intelligent, malevolent magical beast. Exactly what depends on your players' level of course. The drug war is going to take place in the sewers, back alleys, and shady locations throughout the city-state, and it's going to involve humanoids (for the most part).
Most GMs look at that and say they need to build three adventures. I don't. I look at that and say I need to have three different maps (maybe), I need to bookmark some interesting undead creatures; I need to figure out what magical beast - I'm likely to advance it with HD, templates, and/or class levels to get the boss, and use the base creature for the rest of the encouners; and I need a pile of NPCs, most of them likely rogues and fighters. that's six different tools I now have at my disposal going forward:
- A map of a lost desert city
- A map of a mountain region
- A map of the city-state's sewer system, and maybe even of a few businesses in the city-state.
- A list of interesting undead creatures
- An advanced magical beast villain
- A pile of NPCs, and bonus points here if some of those NPCs are tied to locations in my city-state, like a bartender or such.
So my players decide they want to help the fig merchant. They like figs, and he's offered them free figs for a year if they can stop his wagons from getting wrecked by the monster. So now I grab my mountain map, I fill in encounter areas with some of the normal magical beasts, I flip through and find one or two other creatures - assasin vines, whatever - to shake up the encounters, and I have my boss beast for them to defeat at the end. Yay, a fun adventure and free figs for a year!
But I still have other tools for when the players come back. I have two more maps I can use either for what they were originally intended as, or I can recycle them later as something else. The sewers just become a ruined keep, for example. My players don't need to know it was drawn as a sewer - they just want to go kill stuff and take their things. The list of undead is something I can always keep tucked away; they make great encounters in lots of things. I can use the scorpion as a set piece encounter the next time they travel through the desert and come across an oasis.
And the NPCs? Wow, that's a gold mine of stuff for me. I can recycle the stat blocks and make them a different villain, or a merchant, or a gang of thugs they encounter after getting free figs. Or I can just level them up later on when the PCs decide, "Hey, we really should do something about the drug cartel under the city. Everybody's getting high and not eating figs. And if the fig merchant goes out of business, we don't get free figs!" It doesn't matter that they're now three levels higher. Advancing those NPCs is a lot faster than building from scratch. I still have my map and city locations to use.
Later on in the campaign, when the PCs go under the sea to take on the sahaugin who have been raiding ships, I can pull out my lost city map and I'm halfway there for having the adventure done.
Finally, as a GM I let other writers do my work for me. Have a few copies of Dungeon magazine? Then you have a big pile of maps and a big pile of NPCs and monsters. You don't have to run those adventures - swipe the map, tab that cool enchanter to be used later on, take the cleric of Vecna and reflavor him (with a slight alignment change and other small tweaks) to instead be a cleric of a neutral god. There's all sorts of little pieces you can suck up from other sources and save yourself some time; the key, of course, is being able to find it when you need it.
So that's how I give my players the illusion of a sandbox world. I keep several plots available to them and keep recycling the content they don't use later in the campaign.
(As a writer, it's a little harder but can be done; it's basically the same thing but on a smaller scale, along with a few ways for the GM to nudge the players back to the plot you buried in the sandbox.)