I don't know about the trad / neo-trad designations (as I don't tend to put much thought into them)... but I do think it's a difference between just how intricate and involved the "game" part of the roleplaying game is.
D&D Combat is a very intricate "game". All the numbers and all the dice rolls over and over over repeated rounds, numbers going up and down, it's a whole big thing. There's a lot of "strategy" and "tactics" needed to play that D&D Combat "game". And I think there are players out there who would like it if the Exploration and Social pillars had that same sort of intricacy.
But they don't. They usually just default to a die roll or two. That's it. There's no gameplay "tactics" in those pillars... the character just does the action they want or they don't. Follow the gnoll tracks? WIS (Survival) check. Convince the guard? CHA (Persuasion) check. Done and done.
And while I can understand wanting more involved mechanical expressions for those pillars (and goodness knows plenty of people have tried to make them)... I just don't think that's really what this specific game of Dungeons & Dragons 5E is about. It's not about the intricate and tactical dice game for Exploration or Social... it's about following the story. The story takes precedence. Yeah, we still have the intricate and tactical Combat pillar (since that's been the game's foundation from the beginning)... but that's where I think the designers want that intricacy to end.
Building on this a bit, I think the "gameplay" for those pillars is different, and that difference is good for the game overall.
If everything had the mathematical complexity of combat, the game would grind, things would feel samey and monotone. The different pillars of gameplay are different
styles of play, and leverage your bran in different ways. Trying to convince a goblin to surrender instead of fight or trying to navigate a dungeon or a forest
should feel different from slaying a dragon or taking out a group of orcs (those should each feel different, too, I think, but that's not my main point here).
When I'm vibing on the Roleplaying part of the game, my brain is trying to figure out other characters, trying to understand motives, trying to uncover secrets, trying to pick the right thing to say to unlock the truth or to get the creature to relate to me or to influence them. That's a different kind of thinking than "get big number go up." And that diversity helps the combat breathe, helps keep the numerical complexity from overwhelming the game.
Similarly, when I'm vibing on the Exploration part of the game, my brain is thinking about resource management and allocation of time and clearing out the fog of war and asking probing questions about the items and rooms and areas we're finding and how we can arrange our skills as a party to each get something out of the environment (and one of those skills is "make sure nothing bad happens to us."). It's different from both of the other two.
It's been one of my issues with D&D's skill system for probably about half its lifetime. Roll a dice against a DC is a nice resolution mechanism, but it's not the kind of engagement I'm looking for out of Roleplaying or Exploration. Skills have always felt like a bit of a kludge to me, and admission that D&D can't think of any mechanic beyond rolling a dice and adding modifiers, and that everything else is basically off the table.
There are "mechanics" you can use on those pillars. Like, say, you can ask NPC's a number of questions equal to your Charisma bonus that they answer truthfully. That's a gameplay mechanic that would work in a Roleplaying context, when one of the goals of play can be to get intel out of a character. Or, on Exploration, if you use some supplies when you take a Long Rest, that is a mechanic that could add to the experience. But those mechanics don't really involve making skill checks, since d20 + mods is Combat Math, and the mechanics for those other pillars should be different things.