TNG Producer Rick Berman discussed this in an official StarTrek.com interview
Q: Let’s dig into some complicated ground. Roddenberry got sick, became less involved and eventually passed away. What were your thoughts, as the torch was handed on, about following his vision versus doing what needed to be done to make the show work versus any urge you might’ve had to put your own imprint on TNG?
BERMAN: It was never a sense to me of a torch being passed. That all sounds great in retrospect, but things are never quite as clear-cut as that. As the first few years of TNG went on, Bob Justman left the show and Maurice Hurley and I were involved. And then Maurice left and a fellow named Michael Wagner was hired. He lasted a very short time, and then Michael Piller came on. Gene was comfortable with me taking care of the day-to-day supervision of this program that he’d been involved with for about two years at that point, and he stepped back. He’d come to the office every day. He did a lot of correspondence with people. He and I would talk a lot. He’d read some scripts. But his involvement got smaller and smaller as the months went on. Then he got ill and his involvement got quite a bit less. By the time he passed away, I was, I guess you could say, running TNG along with Michael Piller. And I’d been asked by Brandon Tartikoff, at the time, to develop a new show. This was something that I discussed with Gene, who felt very positive about it. But he was quite ill at the time and wasn’t really interested in getting involved with what it was or what it was going to be about. I would like to think that he had faith in both myself and Michael, who I asked to work with me on what became Deep Space Nine.
So, by the time Gene died, there was no sense of “Oh my God, this great responsibility has been put on my shoulders.” I was doing the job I’d been doing for a couple of years and Gene had become, in a sense, a producer emeritus of the organization. I had absolutely no thoughts about putting my own imprint on Star Trek. My interest was to continue to try to do the best work that I could and to hire the best people that I could and to continue on with what Gene set out to do with TNG. It was my hope that the direction we went in with DS9 – and onward with the other shows -- was something he would have thought was the right direction to go. I don’t see myself, nor have I ever seen myself, as a visionary who wanted to put his ideas onto the show. I wanted to be as truthful as I could to Gene’s vision, and that was something I was more than comfortable with.
From Berman's statements, we can extract the following answers to your questions:
Did the post-Roddenberry TNG differ from his initial vision?
No, not according to Berman. "I wanted to be as truthful as I could to Gene's vision."
Did Berman violate any of the rules that Roddenberry had laid out for the series?
Not really. DS9, with its initial lack of a starship, its struggle-on-the-frontier theme, and an imperfect Federation, was probably the biggest departure from Roddenberry's vision. The idea, at least as presented to Roddenberry, did not seem to offend him. "This was something that I discussed with Gene, who felt very positive about it...." (Others have said that Roddenberry was not so enthused, but Berman has always maintained that Roddenberry was supportive of DS9.)
Were there any general themes that became less emphasized after his death? Did Berman make hire or fire decisions contrary to Gene’s known preferences?
Again, no evidence. "My interest was to continue to try to do the best work that I could and to hire the best people that I could and to continue on with what Gene set out to do with TNG."
From another interview with Berman:
"He was a teacher to me in a sense," Berman said. "I was involved in the creation and the ongoing production of the television show that was based on his vision. From that point of view, he was somebody that I looked up to, and in a sense, obeyed, because we were playing by his rules, which was fine by me."
Not all of the "rules" were due to Roddenberry himself
Another point to add is that not all of the "rules" came from Roddenberry himself. William Shatner's TNG documentary Chaos on the Bridge reveals that Leonard Maizlish, Roddenberry's lawyer, and Maurice Hurley, the showrunner picked by Roddenberry for Season 2, had charged themselves with keeping TNG true to the Roddenberry vision and took it to an extreme that Roddenberry himself might not have on his own. From a review of the documentary:
To make matters worse, [Maizlish and Hurley] both pushed Roddenberry’s utopian vision further than even Roddenberry had wanted. Hurley recounts that by the middle of the second season, Hurley was fighting Roddenberry because [he] had gone further with the "no conflict, everything is perfect" ethos than even Roddenberry felt comfortable with.