There has always been a back and forth battle for consumer dollars and attention between the big two comic companies, Marvel and DC. Now that both are building their own cinematic universes, that perceived rivalry has grown to encompass film and TV.
Zack Snyder was asked by FilmInk if Batman V Superman: Dawn of Justice will kickstart a rivalry between the studios, and he had this to say.
“I’m a comic book fan, so it’s a hard thing for me. I have a great interest in the movies that they’re making, and I have nothing bad to say about those movies. But I will say that Warner allow for these amazingly open movies that allow a giant audience. But they’re very careful, they age the movies with the people that are watching them very carefully, in a very interesting and thought out way, but they’re also very filmmaker driven. You can’t imagine those films being made by these very strong filmmakers without a sense of respect and kindness. They say, ‘We hired you to go and make something cool, so show us what that is.’ I don’t know what the Marvel process is. They have their own way of doing things. But I know what happens at Warner, and I feel like the great work that they’ve done here has been based on that philosophy. That will end up being the thing maybe for me which is most rewarding as a filmmaker. They allowed for this unexpected journey for the film, because I was just allowed to make it cool.”
Have to give credit to Snyder for a very mindful and press savvy answer, but in essence, what he is saying is that Warner Bros. tends to make its movies based on the vision and strength of its directors and producers, allowing them to mold the heroes in ways that most aren't afforded. That may be true, but I'm also fairly sure that Joss Whedon was given a great deal of leeway with Avengers, so that may not be an exclusive thing to Warner, though I do see what he is driving at.
As for if this will start a rivalry, who are we kidding, there already is one.