Murray got fired cause he stood by Bylsma, so he had to go.
Here's an article with Pegula discussing in corporate speak why he fired Murray:
What Terry Pegula had to say about firing Sabres coach, GM (transcript)
In the link, Pegula summarized firing Murray for a lack of discipline, structure, and communication.
That sounds about right--Murray had no problem trading for guys like Kane and Bogosian who were not known for their leadership, and in the case of Kane, had a negative reputation around league at the time. He wasn't concerned with off-ice issues as long as it didn't affect on-ice performance.
A big part of it had to have been that Murray was immensely socially incompetent. Some of the stuff he said weren't things you expected to hear from an adult (his "slapping pee pees" comment was probably one of the most wtf moments I've had as a sports fan--not that it was wrong, it just came off like a 14-year-old was running the Sabres.)
Murray was terse at the best of times and his quotes during important moments weren't well thought-out (his comment about losing the McEichel draft lottery, when taken in the context that he was consistent in his view that the worst team should get the top pick was understandable, but repeating it immediately after the draft just set off an unnecessary media-storm that made it sound like he was slighting Eichel.)
I think Pegula just wanted someone more mature... and ended up choosing Botterill, who is as boring as beige but brought his own slew of issues in the competence department.