If i remember correctly, the 2 of them were at an NFL meeting and Jimmy had been drinking a bit and said some mildly unkind things about Jerry and it seemd like the next day, Jerry fired Jimmy.
Is Jones really that thin-skinned?
And Cowboy fan Dopers have any insight?
Jimmy is kind of outspoken in general. And Jerry must think, deep down, that he’s a better coach than anyone he may hire. That’s how he comes across to me anyways.
The whole rift was a culmination of a long, drawn out (and often public) power struggle. IMHO
I’m not a Cowboys fan (shudder), but I’m sure there’s only so much Jimmy Johnson a person can take.
My WAG is that he wanted to distance himself from former owner Tex Schramm. To do this, he had to not only try to coach the team instead of letting the coach do it, but he also had to FIRE his winning coach - ala Al Davis & Art Shell - instead of sticking with him for years and years.
Former cowboys fan here. Which will probably explain my feelings for Jerry Jones. 
What they said.
Total power struggle. Idiot Jerry Jones couldn’t give stand giving up the glory of the Cowboys in their heyday to Jimmy Johnson. Jerry wanted to be recognized as the great mind behind the Cowboy dynasty, but everyone knew it was because of Jimmy Johnson. Proven by the fact that after he left, the almost immediately started sucking. Switzer won a superbow, but it was because of the team that Jimmy Johnson had put together.
Have anyone ever seen the post superbowl video where Jerry practically wrestles the Lombardi Trophy out of anyone’s hands who want to touch it? It’s funny as hell. Jones is such a prick.
E3
Switzer won a superbow,
Laughing as I visualize switzer with a big bow around his head and a pistol in his hand. 
He fired him because he could. That’s why he bought the team – as a plaything to throw his weight around.
It’s also true that those two egos were too big to co-exist in the same space. Both of them shared a lust not only for power, but for self-promotion, and there can only be one publicly-acclaimed mastermind of a team.
Johnson himself represented an increasingly prevalent, and unsavory, trend of coaching: the spin doctor politician as coach. In any given situation, Johnson spent half his time focusing on the team, and the other half focusing on the public image of him and the team. The Machiavellian successfulness of this approach, and of Johnson’s indefatigable courtship of media acclaim, is that he is to this day acclaimed as a genius coach and master psychologist, despite some decidedly mediocre on-field performances.
At OSU, he had medium talent (though it included B. Sanders and D. Manley) and got medium results. At U. Miami, he inherited great talent, did a decent job of recruiting (always he strongest suit), and choked away many of his biggest early games (Fiesta Bowl v. Penn State; Hail Flutie; Sugar Bowl v. Tenn). At Dallas, he inherited the best draft choices ever thanks to the H. Walker trade, and managed again to pick personnel decently (though he did spend top draft picks on Steve Walsh and Russell Maryland – fortunately, he had many more to spare). At the Dolphins, he used four seasons to more or less duplicate the same first-round-playoff-loss results that got Don Shula fired – and lost clutch playoff games in more lopsided/humiliating style than anything Shula ever suffered; all the while being acclaimed as a genius while he successfully convinced the press that injuries/Shula’s playbook/Marino/free agency, i.e., anything but himself, were responsible for the failures.
You’d like to think that Jones began to see through the facade of Johnson’s legend and recognize his limitations as a game-day coach, but I’m afraid he fired him for all the wrong, p_____g contest alpha male reasons.
IMHO it all started when Johnson short-sheeted Jones at the University of Arkansas. 
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No way man; they were rivals back in Arkansas for the affections of Jerry’s sister.