Well, I'm just basing this on the comments I've seen. And I admit, I've always been more of a DC fan than a Marvel fan, so I'm prone to bias on DC's part (though in recent years, I have been a fan of some of their comics, but not their editorial decisions).
With Marvel's long-term success over DC, it always seemed to me that Marvel took the position of the big bully and DC played the part of the weakling getting sand kicked in its face. There were times when the ribbing seemed good-natured, times with it seemed a bit overboard, and times when it seemed downright mean.
Case in point, Jeph Loeb in 2005/6 took some shots at Marvel, especially in his last arc on Superman/Batman
, in which they faced the Maximums (a non-to-subtle rip on The Ultimates
). I remember him making some disparaging/cheap-shots at Marvel. In what seemed no time at all, Loeb quit DC and started working for Marvel. Shortly thereafter, he fired some shots at DC. I just laughed at how quickly his tune changed when the people signing his paychecks changed.
I understand this stuff happens all of the time. Characters/concepts homaged/parodied on both sides, shots fired at both companies, etc. It just always seemed to be more than just good-natured from those like Breevort and he always seemed quick to insult DC. As I said, I may have misjudged him.
It's like sports rivalries. Some fans/teams/owners/players are good-natured about it and keep it up for the fans. Others seem to really hate the other team. And sometimes what seemed good-natured was really hateful and vice versa. At the end of the day, though, it doesn't really matter.