Gary Gygax and Dave Arneson created the hobby together when they wrote D&D. But later on they had a falling out - what caused that?
(We don't need minutiae, just the biggest factors)
Gygax felt that the 'work' side of the whole affair counted far more, but Arneson felt that his 'spark of life' was of paramount importance. The debate continues after their passing; Gygax's fans claim he was cheated, and Arneson's fans claim he was neglected by both history and Gygax.
To understand these conflicting points of view, review the early history:
Dave Arneson contributed crucial concepts which (according to the eventual legal ruling) changed the existing (and common) game form into a unique new one. The previous emphasis on wargaming (with miniatures) changed into roleplaying (with wargame mechanics and terminology, also using miniatures).
Gary Gygax devoted his life to this new type of game. First he collected all the ideas and added lots more of his own. Then he wrote everything down, co-founded Tactical Studies Rules (with Kaye & Blume), and published. He stayed with it through rough early days, wrote even more (Greyhawk etc.), 'rode the tiger' as D&D took off and generated unexpected amounts of revenue, and paid the costs in Stress on himself, his marriage, family, and friends.
Arneson continued his life elsewhere, and did none of the above. Hence, Gygax felt that Arneson was but one of many contributors, and felt that the revenues should go to those who built the company and fueled the D&D 'boom'... himself first and foremost.
In the eventual legal ruling, the "Spark of Life" -- the thing that changed a clever but unremarkable game (the pre-D&D 'Greyhawk' wargame campaign) into a unique and special thing -- came from Arneson. The judge(s) agreed that without Arneson's contribution there would not have been a Dungeons & Dragons game at all.
There have been a number of historical articles written about it (the best one in my opinion is Paul La Farge's Destroy All Monsters—but there are lots of others) But I really think the issues were money, who got credit for what, and the fact that these guys were not located in the same area—distance leads to difficulty in communication, and Gygax was trying to run a business as well as be a creator.
And to top it off: they had different styles, as just as DMs have different styles today. Blackmoor had "holy water fire hoses" and turnstiles outside the dungeon—he had a very whimsical style. Gygax had his own "Greyhawkian" style.
Also, it's really hard to maintain long, creative, collaborative relationships when you have two especially talented people. You might as well ask what drove Lennon and McCartney apart.
At the end of the story, I find myself doubting there was anything specific that caused a falling out other than the situation itself.
When I asked Dave about it, He cited creative differences and Gary stealing all the credit as major issues, but fundamentally, it was Gary trying to avoid paying him royalties that was the real rub. That was back in 2004 or so.
My impression is that the two gentleman had different attitudes and goals towards the market and Dungeons & Dragons. Gygax wanted to run a game company while Arneson was more laid back. The two attitude didn't mesh. You can read some of what went on starting in this Dragonsfoot thread. Mind you it from the point of view of Tim Kask and Gary Gygax.
After reading this and other accounts the whole thing strikes me simply as two people with two different attitudes towards the game and what to do with it. Gygax wanted to do what it took to make D&D a commercial success, Arneson was more laid back. Neither was wrong but they were incompatible attitudes and lead to later conflicts about division of royalties. And years later they reconciled and went their separate paths.
Some good points have been made. I would say it came down to credit, and direction. Credit, in that for years there were arguments about who did what first, with both sides having their own followers and adherents. Direction, in that EGG was obviously motivated for a more commercial product; while Dave didn't seem to be pushed in that direction (besides First Fantasy Campaign, and Adventures in Fantasy, Dave really never released anything RPG-wise on his own until a few years before his death). Plus if you read up on the history of the game there is never much evidence both these guys were bosom buddies to begin with, so it's not like it was a huge stretch to imagine they were looking at their creation from totally different directions and got cross-wise.
I talked about this some with Jon Peterson of Playing at the World. His characteristically primary-source-driven perspective - hopefully not distorted by my hearsay account - was that, during the months Arneson was employed at TSR, you can see him putting out regular issues of his Napoleonic fanzine (presumably composed on company time and produced using company equipment), while it's hard to identify things that were his unique contributions to TSR's work at the time. Jon's take was that Arneson wasn't all that into this fantasy thing; it was something he'd dreamed up that others became more enthusiastic about, just like (according to David Wesely) Dave Megarry became very excited about the idea of delimited choices in a dungeon environment during the first Blackmoor session that took place in an underground setting and ran with that ball (in Dungeons of Pasha Cada, later to become Dungeon!) in a way Arneson might not have.
However Playing at the World did make me realize that Gygax and Arneson were in cahoots for longer than I'd thought. They both seem to have been the most prolific and ambitious members of their respective scenes, and in addition to their collaboration on Don't Give Up the Ship and Arneson's use of the Gygax/Perren Chainmail rules for Blackmoor combat, Dave also used Gary's Diplomacy variant to handle the strategic level of that Napoleonics campaign, in which Gary (and other Lake Geneva folks) were regular players.