Since everyone's speculating now.
Here's why I've always thought was their biggest difference of opinion.
I believe that Kevin Pritchard doesn't think Nate McMillan is the Coach who can get this team into the playoffs. I think that Kevin has a very different philosophy for what it takes to win in the NBA than Nate McMillan. (Kevin likes guys like Batum and Bayless and Rudy, who can put pressure on the team's ball-handler, is smooth in traffic, can get to the hoop and get to the free-throw line, where they might turn the ball over a few times, but the complete pressure they put on their opponents outweighs the turnovers. Where, Nate likes guys who stand around, shoot the three, don't take risks. One type of basketball is fun to watch, while the other is boring.)
Paul Allen loves Nate McMillan, and when Kevin Pritchard's brought up the idea of getting another Coach, there was a clash there. Nate didn't play Bayless and Batum as much as Pritchard wanted him to, and this probably came up at meetings.
One of these two guys had to go....McMillan or Pritchard. I always thought, and hoped it would be McMillan, but it was Pritchard, and the reason is that Allen loves McMillan.
As "reported", I don't think it has anything to do with "who gets the credit", as there's no credit to be gotten. We still haven't won a playoff series.
Kevin Pritchard was fired because he wanted Nate McMillan replaced, and instead of replacing McMillan, Paul Allen fired Pritchard.
You can try to discount this idea, but I feel I'm 100% correct about this.