Saving Throw = Defy Danger

OSR = trusting the GM to improvise gameplay resolution on the fly
PbtA games = trusting the players to introduce plot on the fly

This is the only difference I really see between these two kinds of games.

In Dungeon Crawl Classics (an Old School Renaissance game) each character starts with a random profession. The rules for skills barely fills up two pages, but the idea of this received profession is that your character can make impromptu skill checks related to the profession. There is no real rule for this and arbitration is left up to the GM's fancy. The idea behind this is that if one of Greg's characters was a wainwright, then he could conceivably know everything there is about carts and wagons and the construction thereof, and if confronted with a trap or contraption in a dungeon that uses a cart then he could potentially roll his Intelligence with a suitable bonus to figure out how it works. Or something similar, dependent upon profession and situations arising that might relate. For example, in the last game of DCC I ran one of the characters was a miller, and the player wanted to use the character's knowledge of how a grindwheel works in a millhouse in order to bypass some scenario.
Improvising gameplay resolution.

In Dungeon World (a Powered by the Apocalypse game) each character has access to a "move" that acts as an all-purpose skill check for your character that plays off of Intelligence called Spout Lore. Make the roll and your character knows something interesting and useful, make a partial success on the role and they only know something interesting, it's up to them to make it useful. So let's say Greg is playing a druid and uses Spout Lore to know what kinds of animals live in the region, it doesn't matter how well they did just that they didn't miss the roll, and the GM can't think of anything, and asks Greg "What are you hoping to find in these woods?" and the player says "Something that seems out of place, like venomous snakes." The GM says "Okay, sure, there are poisonous snakes infesting these woods, nobody knows where they came from." and just like that an interesting plot point has been created. Not very useful, but maybe the GM tells Greg the best way of catching one of the snakes without getting bit. What started as a player looking in the wrong direction for a clue to their current quest ends up being the seed for a potentially all-new quest, then it's up to the players whether they investigate these snakes.
Introducing plot.

These are, of course, really generic examples, and they have to be in order to express how these systems work. But there's another rule dynamic where this difference-which-is-actually-a-similarity can be highlighted, OSR gamers call it the saving throw. There are some OSR modules have so many saving throws that sometimes I think that those are the only stats worth leveling up. Dungeon World, and Apocalypse World, have their own rule for the saving throw called Defy Danger, and Acting Under Fire.

A saving throw is best described as a roll one makes for their character in order to avoid some kind of effect or hazard. Compare the saving throw with Defy Danger which is literally described as "When you act despite an imminent threat or suffer a calamity, say how you deal with it and roll." Compare this also with Acting Under Fire which is described as "Call for this move whenever someone does something requiring unusual discipline, resolve, endurance or care." In the environment of a game, one of these rules is just a mechanic that allows the GM to call for a roll to avoid some type of hazard and the other is the exact same thing!

In an OSR game a saving throw is pretty straightforward, the player rolls the dice and if they hit their save they either avoid or lessen the effect of whatever danger they're exposed to. Sometimes the number they need to roll is known, sometimes it's secret; they almost always need to roll high. Resolution of Acting Under Fire or Defy Danger works similarly, but here's that partial success area of the roll, the 7 to 9 range, if the roll comes up in that range the GM will give the player a choice, usually both involve succeeding with some kind of cost attached, and then the player decides what cost their character pays. Depending on the situation, this can change the story.
And that's the only difference.