Takei has outright said, repeatedly, that Shatner was a prick. Not line hogging or bad directing. That he was actively a shitty person to be around in any form and that he didn’t respect anyone around him. This is backed up by his having footage made of someone for a movie. Someone who, ahead of time, said he did not consent to the footage.
Just because other people are pricks does not give him a pass to be one. Plenty of people work in positions just like him and aren’t nightmares to work with.
That some kind of falling out is what I just described above. Shatner was making a Trek documentary and asked Nimoy to be a part of it. Nimoy categorically said no. Shatner then discretely recorded footage of Nimoy anyway, against his consent, and used that footage in his movie. Nimoy was furious and never spoke to Shatner again. Shatner then wrote a book about his friendship with Nimoy where his name was almost as big on the cover as Nimoys. In the book he admitted to doing this and them not talking.
In that same book, Shatner demonstrates repeatedly why he’s an asshole by such things as getting angry that Spock got more fan mail than him.l and says the only reason they had any basis of a friendship was because Shatner saw Nimoy be a hard ass with contract negotiations so they started negotiating together.