TL;DR - The difference was scope: Dragon(broad) vs Dungeon(narrow)
Dragon was a gamer's magazine (wargames and RPG's mostly, and certainly D&D heavy) that covered games from a variety of publishesr. Dungeon was a TSR RPG adventure magazine.
Dragon began as a gaming magazine that covered far more than D&D
In number one, the first feature article was The Battle of Five Armies played in miniature using Chainmail rules. Most of the rest was D&D centric, plus a short story and a listing of gamer conventions. Issue two declared Dragon to be "The magazine of Fantasy, Swords and Sorcery, and Science Fiction Gaming" on the cover, contained two swords and sorcery fictional short stories, ads for historical miniatures, a review of the game Venerable Destruction, and D&D material. The third issue had a science fiction themed cover, a review of the SF game War of the Empires (play by mail), more fiction, Finieous Fingers' debut, an article on good miniatures of all sorts, and D&D stuff. Issue 4 had massive treatment of the Empire of the Petal Throne, Metamorphosis Alpha notes (SF game) from James Ward, and a bit of D&D.
Over time, Dragon included articles, editorials, fiction, games(the first was Snit Smashing), small modules, designer commentary, reader letters, and hosts of unofficial ideas for new classes, spells, and more. Game reviews featured prominently; reviews for all kinds of games. It began and stayed a gamers' magazine, in the broad sense, though D&D material was certainly featured in most issues.
- I have a 1988 issue that reviews The Bards Tale by Electronic Arts, an Atari game. In that same issue, there's an ad for Runequest, GURPS, a supplemental article for Runequest, and a play by mail game called It's a Crime).
- I still have my copy of Citadel by the Sea, a Sid Fisher module from Dragon 78 that I kept in a file folder (I've run it twice; but the magazine is long gone).
Like its immediate predecessor Strategic Review, Dragon covered more than D & D. Tim Kask (Dragon's first editor) pointed out in an early issue that it wasn't a TSR house organ; ads and articles for other games were to be featured and welcomed with open arms1. It replaced the Strategic Review because the gaming hobby fan base demonstrated to the folks at TSR that there was a demand for such a magazine. (For comparison to a contemporary (even rival) publication, see Avalon Hill's "The General" magazine that also supported the larger wargaming hobby (my two copies of that mag are long since gone ...)).
Strategic Review had also covered more than D&D. It had information on miniature combat, naval combat, micro armor, arms and armor, and material for Boot Hill (Wild West RPG).
As Dragon and the hobby matured, a variety of material that was later published (for example, the Barbarian class that ended up in 1985's Unearthed Arcana) got its first printed form as a new feature in Dragon #68. (Thanks @Kirt)
Dungeon was more narrowly focused: adventures for TSR RPG's.
As the RPG hobby grew, the customer demand for more material, and a chance to show off their own adventures, led to TSR splitting the adventure/game feature from Dragon into its own publication. (Dragon was getting up in page count by then ...) Dungeon hosted both professional and amateur modules. (Dungeon #53 has Elexa's Endeavor, by Christopher Perkins).
On the cover, under the word Dungeon, it says "Adventures for TSR Role-Playing Games." This was from its inception a TSR house organ. From Roger E. Moore's "Out of the Dungeon, Into the Fire" editorial introduction to that magazine:
DUNGEON Adventures is a new periodical from TSR; Inc., in which you,
the readers, may share your own adventures and scenarios from AD&D®
and D&D® gaming with the legions of other fantasy gamers. Each issue
offers a number of fairly short (but often quite complicated and
long-playing) modules, selected from the best we receive. What kind
of adventures do you want to see? Were going to offer as broad a
spectrum of material as possible: dungeon crawls, wilderness
camp-outs, Oriental Adventures modules, solo quests, tournament
designs, BATTLESYSTEM scenarios, and more. Of course, what we have
to offer depends on what you send to us. (See our guidelines offer on
page 60.) Write in and tell us what you want.
Contrast that with Tim Kask's initial efforts to declare that Dragon was not simply a TSR house organ ... From Dragon Rumbles, in issue number 1 of Dragon
This issue marks a major step for TSR Hobbies Inc. With it, we have bid farewell to the safe, secure world of the house organ, and have entered the arena of competitive magazine publishing. We have activated a new division of the corporation; TSR Periodicals. We are soliciting advertisers, and giving notice to the rest of the pack that we have arrived with a vengeance, with a mission to fulfill. That mission is to publish the best magazine devoted to Sword & Sorcery, Fantasy, Science Fiction and Role Playing gaming. It is a newly developed field, and we admit to being only as old as the following. However, we feel that our experience, gained from publishing the STRATEGIC REVIEW, the pioneer in the field, will stand us in good stead. But, we also make allowance for the fact that as the field grows and expands, demands and needs will change. We actively encourage your suggestions, criticisms, or whatever. We can adequately serve you, the gamer/reader, only if you let us know what you like/want. We plan to include gaming, variants, discussion, fiction by authors both known and unknown, reviews of interest to our readers and anything else there is a demand for.
Dragon made good on that promise.
As I read it at the time (and as Moore's intro confirms), one of Dungeon's goals was to attract high quality adventure content. (Neither of the two modules I submitted back then made the cut😕). It was more limited in scope than Dragon, with the module/adventure being the core product people bought it for, to use in their games, or at their local club conventions, etc.
1 From Dragon #5, p. 3
I have extended invitations to a number of authors of fantasy and
science fiction games, other than D&D and EPT, to write on their
creations for these pages. While we recognize that D&D started the
fantasy gaming genre, there are now a number of science fiction and
fantasy games available that we feel should be treated in this
magazine. I extend this invitation to non-authors (of games) to do
this also. I’m looking for articles on STELLAR CONQUEST, THE YTHRI,
Contents WBRM, GODSFIRE, STARSHIP TROOPERS, OUTREACH, SORCERER,
STARSOLDIER, GREEN PLANET TRILOGY, OGRE, MONSTERS-MONSTERS, VENERABLE
DESTRUCTION and others. It’s time for THE DRAGON to expand its subject
matter. I want to get into fantasy miniatures as well.