I didn't know that, so I dug around for the cause. From a recent NYP article:
Takei married his husband in 2008 and Shatner became enraged when
he claimed he wasn’t invited. He ranted against Takei in an
online interview. [...]
Shatner called Takei’s wedding a publicity stunt and hypothesized
that their feud stemmed from Takei’s unwillingness to playing
second fiddle to Shatner on “Star Trek.”
It seems to be mostly based on petty, even comical stuff like this. This may sound paradoxical, but as someone who always had a lot of respect for both of them, this is disappointing and a bit of a relief at the same time.
That wasn't the cause. According to Takei, they didn't get along even during Star Trek. "Petty, comical stuff" is what the media narrative consists of because they're two amusing personalities, and we want to be amused by them. Also they're two old men who wouldn't have spoken for the last 50 years if it weren't for an avalanche of fan-cash that has forced them into continued contact and some civility.
From what I can tell he's had a bad relationship with a number of the other cast members since the original series's filming, since they thought he was too self-centered and kept trying to minimize screentime given to the supporting cast. The feud became kind of well known among fans, because four of the actors (Walter Koenig, James Doohan, Nichelle Nicols, and Takei) often aired their dislike of Shatner at Star Trek conventions. Though for his part Shatner claims it was a one-sided feud he didn't even know about until 1993: he says he discovered the issue when he tried to interview a bunch of the old cast for his memoir, and several wouldn't talk to him, or did but had negative things to say.
>Though for his part Shatner claims it was a one-sided feud he didn't even know about until 1993
Which, of course, humorously confirms all of the casts' characterizations of him as a selfish egomaniac in a wig and a girdle. I think that Shatner is aware of this (within the past 20 years, not at the time) and plays into it for the humor value and a media bump if he's currently selling something. He's always selling something.
