It's been a long time since I read the book but as I remember it:
- Lisa was a big-time fancy project with established engineers. They didn't like Steve's style, so (basically) they booted him from the team.
- Steve moved to the Macintosh team, which had more of a startup feel. (Think 'Bad News Bears')
- In 1985, Steve tried to force the company to fire someone and got himself fired instead.
- Steve started a new company called NEXT and worked some other great projects like Pixar.
- He learned to be a little less abrasive.
- In 1996, Apple got him to come back and run the company again.
- iMac, iPod, iPhone, iPad...
One update. He came back to Apple because Apple acquired NEXT. Apple desperately needed a new OS after several failed attempts to replace the original Mac OS. They bought NEXT for that purpose. The current OS X is a direct decedent of NextStep.
When Jobs came back to Apple it was initially downplayed, that he was only there as an adviser. I think everyone knew what was really happening. I think the fake drama between him and the then CEO in the movie was just that, fake drama. If Gil Amelio didn't know what was coming then he didn't deserve to be a CEO of a major corporation.
Amelio was taken in by Jobs and completely fooled. After the NEXT acquisition, Jobs was extremely coy about his intentions/future plans. Initially he came on as an adviser to Amelio.
If you read Walter Issaacson's biography of Jobs, this part of his life is covered extensively (and it's not sugar coated). At the time, Jobs was CEO of Pixar and enjoyed running that company. He said initially he wasn't interested in taking over operations at Apple because it was such an enormous task (the company was in serious trouble - at at one point was nearly insolvent). Apple had a lot of problems and Jobs was not an experienced CEO (at Pixar, he provided financing and negotiated distribution deals with Disney) in the kind of hard core/cutback management the company needed at the time.
It's true Jobs worked behind Amelio's back to oust him, but it wasn't entirely about himself taking over. In addition, the board of directors realized the only hope the company had was to bring Jobs back on in a key role
It wasn't just the board. All of us in the Valley knew too that Apple depended on his returning if they were going to ever recover.