There is no difference between a module and an adventure. Module is simply an archaic term for adventure.
Early TSR D&D/AD&D adventure products called themselves "modules," but in their covers and text made no distinction between an "adventure", a "module," and an "adventure module," they used the term interchangeably.
Here's a picture of X1 Isle of Dread from 1981. The header claims it's "Dungeon Module X1." The subheader declares it "A Wilderness Adventure." The text says "first in a series of adventure modules." The oval at the bottom calls it a "Special Introductory Wilderness Module."
"Books - Modules - Miniatures, the Dragonlance Saga Continues," proclaims the back of DL3 Dragons of Hope.
Other publishers didn't adopt the "module" nomenclature as much. On my various non-licensed Mayfair Role Aids AD&D adventures and sourcebooks from the 1980s, they studiously just call themselves "adventures."
But are modules setting independent?
Many were, of course, with the idea that you could use them in your own campaign world. But there were many modules explicitly set in Greyhawk or another published game world. T1-4 Temple of Elemental Evil proclaims on its cover "A Monstrous Adventure! Includes Tl , The Village of Hommlett and the long awaited T2, Temple of Elemental Evil set in the WORLD OF GREYHAWK Setting." There was also a WG series of modules that was not just set in Greyhawk but got deeply into the locations, NPCs, and metaplots of the setting beginning in 1982.
But are modules more - "modular", and not full campaigns like an Adventure Path is?
Not by the 1980s.
There was no term "Adventure Path" back then, but when they put together larger arcs of adventures they were still called modules. Take the plot arcs like I3-I5, Desert of Desolation series. When republished, the foreword talks about "[the authors] provided an excellent set of adventures in the original Desert of Desolation Series, I3 through I5. The story that held these adventures together was of a campaign style, an epic quest against seemingly insuperable odds." It was recognized that you could have series of adventure modules that did provide an entire campaign.
Take D3 Vault of the Drow (1980) - its cover says "This module can be played alone, as the conclusion to module D1-2 (Descent Into The Depths Of The Earth), or as the third module in a series that forms a special extended adventure (G1-2-3 Against the Giants, D1-2, and Q1 Queen of the Demonweb Pits). (Eventually it was republished as GDQ1-7 as one large softcover.)
Or T1-4, Temple of Elemental Evil, which when published in one larger softcover, its cover refers to it as "Campaign Adventure," but then in the Introduction starts in with "This module is designed for..." Modules could have plot and be interlinked and have many parts.
In fact, I can't find the term on any of the products themselves, but when the original combination modules (T1-4, A1-4, GDQ1-7, I3-5) came out, we called them "supermodules." In fact, that has to be somewhat official because it's in the product name on Amazon!
TSR continued to (somewhat intermittently) use the term “module” through 2e in the 1990s, but left the terminology behind with the advent of 3e.
Basically, "module" was a term TSR started using early in D&D, probably because earlier wargames had "expansion modules", and it stuck for a decade, but then gave way to the more readily understandable "adventure", especially as other publishers who didn't start with that term published games and supplements.
All "well to me adventure means this and module means this" statements are basically made up out of people's minds, except inasmuch as "module" reminds one of 1970s-1990s D&D.
Nowadays, people use "adventure" and "adventure path" as common terms, but really Paizo coined the "adventure path" nomenclature in the late days of Dungeon, before then you just had "campaigns" or "campaign adventures."