Was the project re-started with new genesis block but with the same name?
No, the project was never restarted with a new genesis block. The project simply continued with a new set of stewards of the project. These stewards replaced, or more bluntly put kicked out, the old team, which was led by thankful_for_today. That is, after some discussion on IRC, the "move" / "takeover", in conjunction with a name change, was announced by David Latapie on the Bitcointalk forum. The stewards, better known as the core-team, that replaced thankful_for_today were fluffypony, tacotime, NoodleDoodle, smooth, othe, David Latapie, and eizh. At the end of 2015, the latter two core-team members resigned from the core-team due to time constraints and were replaced by luigi1111 and ArticMine.
Was the code just forked?
As far as I can see from the Github commit history, the stewards indeed forked the code over to a new Github repository and continued with that. The old repository of thankful_for_today can be found here.
What was the difference and disagreement?
This basically boils down to thankful_for_today playing the benevolent dictator and pushing changes which the community largely disagreed with. He also pushed changes without conferring with, or simply ignoring, the community. There are more extensive examples in the formal write up, under "the second renaissance".
What exactly happened at this point in history?
There's a more elaborate and formal write up here. This write up also includes some kind of timeline.