To me, the main thing that makes Jerry different from other owners (and each owner is going to have their unique qualities) is how infrequently he has changed front office leadership.
He hired Krause shortly after buying the Bulls and left him in that position for 18 years. In fairness, his initial patience with Krause paid off massively and he didn't give Krause a ridiculously long leash post-MJ. Then the GarPax era begins which most of us feel lasted at least a little and perhaps a lot too long. It did end at a point where it seemed impossible that it could keep going, though. So he's really just on his third front office for the Bulls after almost 40 years of ownership. Again, hard to complain about keeping Krause for a long time given the team's success. His patience with GarPax didn't really pay off although there was a period in which the Bulls had good, likable, and fun teams that came after a long rebuild period. I suspect this colored Reinsdorf's view of what you need to do in order to get a successful team built.
He buys the Sox in 1981 with Roland Hemond as the longtime GM. He leaves Hemond who finishes building the highly success 1983 squad. The next two years are basically an unmitigated disaster so Jerry decides to hire his own man. So far so good. Well, his own man is Hawk Harrelson. Oops. At least that mistake wasn't allowed to go on for too long (I've never found it too clear whether Hawk really left on his own accord as he said publicly or was just allowed to say that to save face). Then Larry Himes takes over for 4-5 years and does a quite good job but Jerry doesn't keep him due to unclear reasons, some mixture of Himes being too involved in day-to-day running of the clubhouse and Jerry being too involved in the duties of the GM. Himes has a not-very-successful run as Cubs GM afterwards. Then, Ron Schueler takes over and builds some decent teams around core players largely acquired by Himes, culminating in the 2000 playoff season. Schueler steps down due to his age as best as I can tell, thus starting the 22+ year KW/Hahn era. Jerry surely feels validated that he gets a World Series in KW's 5th year on the job, validating a bit of patience as well as hiring from within when you're happy with the outgoing GM.
There was some turnover early with the Sox, but I think Jerry must have seen that as tumultuous and something to be avoided in the future. I think he would have happily retained Schueler if Schueler wanted to keep the job — and why not, considering Schueler had just assembled a young team that won its division.
As a member of the group of owners, JR has earned his rep for being tight. But as an owner, I don't really see him as a huge tightass anymore. The Sox spend in a way that seems consistent with where they stack up to other teams as far as market and potential revenue. Sox have avoided big contracts in the Kenny+Hahn era, but I can't help but wonder how much of that is driven by Jerry vs. Hahn in recent years. The payrolls have certainly been high enough to carry the contracts of some of the high-profile players we struck out on so it's not as if JR is unwilling to get near $200M in payroll. Sox used to not spend much in international free agency or the draft, but AFAIK since those categories of spending got standardized we have been spending basically every cent we are allowed to.
In his older years it has seemed to me that JR has embraced the idea of the owner's role being someone that hires a GM and hands over a budget; great, but now there's even more scrutiny on who that GM is. I strongly suspect Hahn is headed out the door, and I'd be surprised if Kenny doesn't do whatever Hahn does. Will it be a firing? Probably won't be described as such but I don't know. Highly unlikely that JR lives long enough for us to know whether he would give the next GM too long of a leash.