Gygax and Arneson, the renowned wargamers and later role-players, developed two different types of games after a while. Gygaxian play [1, 2] is known for being procedural, more rules-heavy and strict; while Arnesonian play [3] is known for being rules-light, inspiring the FKR [4, 5] movement.
There's a lot of historical nuance built the above statements. For example, Gygax was not always such a rules-lawyer, but had a phase of it [6]. Also, the influence of Arneson's games on FKR is hotly contested and perhaps not even encompassing for all FKR games. However, I bring up this glossed-over history as a way to introduce metonymic terminology: the Gygaxian game and the Arnesonian game. They are what you might expect from the above descriptions:
Gygaxian game: Games in which there are rules and procedures to facilitate action.
Arnesonian game: Games which rely more on common sense for determining outcomes of actions, and use dice rolls (though it may not be tied to any specific mechanic or game stat) to determine outcomes of actions.
A first note on these definitions is that they are personal: I came up with the wording based on my experience with OSR and FKR discussion. They are not historical, nor are they meant to be prescriptive. Their descriptive ability is also dubious: categorifying a given game (either rules or campaign) may be undecidable depending on the game. In fact, a given game may include elements of both (and often do).
A second note: they are not directly opposed. A better term might be competing. Certainly in the (OSR) game community as a whole, procedural vs. freeform play are two sides people stick to and argue for over the other. They may not be directly opposed, but also they seem mutually exclusive. If you have a consistent dice roll to resolve a given action, then it leans more Gygaxian. If you think about how a situation would play out in the game world, it leans more Arnesonain. They are a spectrum, and game styles to take ideas from, rather than side with.
A third note: it is totally fine to find yourself on one side of these styles than the other. Everyone has preferences.
An interesting thing about the historical Gygax and Arneson games is that they played very similar games: dungeon-delving, treasure hunting games. The games, in their game world, are exactly what D&D intends for gameplay, and the in-world action of each is basically indistinguishable. This might run counter to what many people think: that you need strict procedures to run a dungeon game and should allow a rules-loose approach to more story or character-driven games. Granted, there are some differences in the dungeon design going on (naturalist vs. mythic underworld), though that seems something a referee or players might prefer separately from rules approach. But in any case both can, and have, worked for dungeon games.
There also seem to be trends in the actual played games on either side of the two styles beyond what is stated above (though I stand by the above as being the essential difference between them). In Gygaxian games, characters amass power through levels and advancement. However, in Arnesonian games, which may lack a numerical and mechanical advancement, you might see changes to characters that is not quantifiable, but may be succinctly described (compare to the notion of foreground growth [7]). These are design choices that often mesh well with the above ideas, but of course are not inherently tied to them.
Gygaxian play gets a lot more attention. I attribute this to a rules-forward approach allowing quick onboarding and cross-table comprehensibility, as well as rules being an easy language for many other analog gamers to understand. Although, there is no reason the world-first approach of Arnesonian gaming cannot fulfill this. It should be as easy as telling someone what the known facts of the game world are. It doesn't seem to have caught on, though. It also may be tied to a desire for engineered success.
If I were to make suggestions (which I am not in the habit of doing), I think the Arnesonian gameplay can be elevated. Thinking adventure-first is a good way to start. Changing numbers to qualities is a good way to free up what a thing does. Thinking beyond mechanics of an encounter to motivations and strategies of dungeon inhabitants gets you on the right path. Giving tools, rather than upgrades, is key.
Modern OSR has fragmented in design choices so much that it takes various aspects from both. What's overall important is to find something that gets your imagination going and flows at the table.