Hand-in-hand with OSR comes the concept of "sandbox play," a style in which the DM does not create scenes or story beats, but rather maps out locations and creatures/NPCs, gives them goals, and starts them rolling, then turns to the players and says, "What do you do?" There is no story until the players bring one to the table; what scenes or exciting things happen are purely emergent based on what the players do.
The Lost Mine of Phandelver in the Starter Set has been largely praised by reviewers for its "sandbox" nature, especially the portions that take place in and around the main town. There are multiple potential patrons with sometimes conflicting goals, and there are multiple ways to get involved with and approach most of the adventure locales. There's only one real "railroad" moment, and that's right at the start of the game: you will be ambushed by goblins as the first encounter. From there, even though there are adventure hooks, you don't have to follow any of them and it isn't assumed that you necessarily will.
Some of the hooks are obvious: your patron has been carried off by goblins and if you want to get paid (or are simply loyal to him), you'll probably want to go track him down. But the adventure doesn't break if you don't. Once you get to the town, there are plenty of other factions to get involved with or adventure leads to follow up on. For that matter, there are trails leading out into the wildnerness, so you don't even need to go looking for adventure hooks if you don't want to. You can just head out on the road and see where it leads you. Of course, if your initial patron dies, you'll lose the benefits of having him around and any further leads he might have had for you, so it's not without consequences– but it's also not a "game over" screen, so to speak.
Prior to Dragonlance, this was actually the norm in D&D adventure design, and in some ways it's very liberating, for both the players and the DM. In a story-based game, the DM has to make sure there are no major plot holes, or the players will immediately and inevitably find them and break your story. And you have to be sending the players through a story they're interested in, or else the whole thing will fall flat at best, or create friction at worst. As a player, I've spent sessions grinding my teeth because I felt forced into a scenario that I didn't want to participate in and had no control over, because there was a plot I was supposed to follow whether I wanted to or not. As a DM I have certainly been guilty of forcing that on my players in the past as well, and I always regret it afterwards.
But it's not like it's all "story-based bad, sandbox good." One main pitfall of a sandbox game is the chance that when you ask the players, "What do you want to do?" they'll shrug and say, "I dunno, what do you want us to do?" I recently encountered an extreme version of this with my Eberron game when I presented the players with a list of jobs available at the adventurer's guild, asked them to pick one, and they simply stared at me. It was not unlike trying to run a campaign based on Bartleby the Scrivener, and I'm still trying to figure out what I did wrong there.
The other major pitfall, from what I've read, is that the players will feel like there's "nothing to do." They might hear a rumor of a dungeon across the mountains or a shipwreck on an island, or perhaps they're even wandering from wilderness hex to wilderness hex having a long string of random encounters, but none of it feels like it matters. "When do we get to the story?" seems to be the chief complaint of players in this kind of situation, to which the standard sandbox answer is, "There isn't a story, until you make one."
Right now at least, as a DM I'm leaning towards the sandbox model. It requires a lot of mapping out locations and writing up encounters that may or may not be used, but on the other hand, I don't have to keep coming up with a never-ending stream of plot twists and compelling narratives. I once had a player flat out tell me, "I don't want to make a story, that's your job." At the time I didn't know what to say to that; these days my answer would be, "Why should I have to do all the work?"
...Which bring me back to 5E, and the bounded accuracy model. 3.x/Pathfinder, with its extreme power scaling, could be run sandbox style, but wasn't great at it. An encounter that would be a TPK at one level, would be a cakewalk two levels later, and the whole narrative flow of the game, as well as advancement and treasure acquisition, was based on the model of "mostly normal encounters, plus one or two challenging ones and one or two easy ones." That meant that you had to constantly scale the world up to match your group, or at the very least make sure everything was in a fairly narrow range.
For sandbox play, that pretty much sucked, because it meant constantly retooling the world around the PCs. This was usually done by moving them from zone to zone like an MMO, so characters didn't start to wonder why, when they wiped out that cave full of goblins, it was replaced with a cave full of trolls.
Theoretically at least, with 5E's flatter power curve, the basic ecology of a region can stay the same and still have interesting or challenging encounters over the course of several levels. The wilderness encounter table in Phandelver, for instance, has something as piddly as three stirges (75 XP) all the way up to something as fearsome as five ghouls (1000 XP), and is intended to cover levels 1-5. I pity the group of 1st level characters who get set upon by five ghouls in the middle of the night– but the possibility of that kind of thing happening is a hallmark of both sandbox play, and OSR. It's also something that you probably wouldn't see in 3.x/Pathfinder.
 Or 4E either, I'd imagine, but that's because 4E would want to set it all up on a map with a giant magic boulder rolling around in circles doing necrotic damage every other round for no good reason...