While the fan-favorite JLA/Avengers crossover wasn't afraid to point out the major flaws in each hero's world - and in both companies as well.
The very first Marvel/DC crossover, Superman vs. The Amazing Spider-Man, was released in the 70s and sold well enough to prove the concept was worth investigating further. A Batman/Hulk crossover only cemented the idea in executives' heads, and plans to create an epic Justice League/Avengers crossover moved forward. Unfortunately, difficulties in communication combined with the companies' different work styles resulted in multiple delays, concluding with the book's cancellation. 1996 would bring Marvel vs. DC - a multi-company crossover, but not the adventure the writers intended.
While the 1996 crossover forced fights between characters (Hulk and Superman fought simply because they were teleported to the Grand Canyon, for example), 2003's JLA/Avengers actually showcases a true ideological difference between the teams. When the Justice League find themselves in the Marvel Universe, they are shocked at state of the world; dictatorships like Doctor Doom's Latveria and poor nations like Genosha are seemingly without help. Superman in particular is incensed, wondering why the Avengers aren't doing their job - especially when he finds out the general public doesn't always their costumed crimefighters. The Avengers, upon journeying to the DC Universe, come across the complete opposite problem.
The Avengers are astounded to discover that the Justice League is celebrated; Superman receives medals at the UN, Wonder Woman is a spokesperson of the World Health Organization, and even the Flash has his own museum. "Look around - this is their city. It wasn't built by men. They must own this world," believes Captain America. "Like little tin gods - demanding the public's adoration instead of protecting its freedoms!" This is the core difference between Marvel and DC: how the public reacts to their heroes. Of course, the reason why DC civilians adore the Justice League is because they are allowed to change the world, while the Avengers are not - and this isn't due to supervillains, but Marvel writers.
Marvel sets their stories in the real world - cities are not created out of thin air like Metropolis or Gotham, for example. Unfortunately, this means real-world atrocities - 9/11, the US invasion of Iraq, etc. - must take place...which means the heroes are not allowed to prevent them. Thus, Marvel fails to allow their heroes to be heroic, while DC's citizens fail to be realistic; this doesn't detract from their stories in the slightest, however, and the Avengers and the Justice League are still the paragons in their respective universes.