I started to think over the question. I don't think there is really any fundamental difference with regards to the actual heroes. Characters like Barry Allen and Peter Parker, or Captain America and Superman, are pretty good equivalents of each other from an aspect of personality and morality within their universes.
What I believe is the real difference between Marvel and DC is that with Marvel, the heroes are constantly having to fight the society around them. Peter Parker has J Jonah Jameson using his newspaper to paint Spiderman as a villain, while depending on him for his livelihood. Bruce Banner is hunted by the military as Hulk. The X-Men are feared and persecuted as mutants. The Punisher is wanted as a criminal. And so on.
In comparison, DC is assuming a society which largely appreciates their heroes. The GCPD is to a degree ambivalent about Batman, but then it is also explained that it is the corrupt parts of it that are against him. Superman is held up as a symbol of hope. The Justice League acts openly in the world. It does happen that they have to deal with a belligerent surrounding society, but it's either the result of enemy action (like in the case of Maxwell Lord) or it's the clearly set up as an exception (like in Frank Miller's The Dark Knight Returns), but even then they usually see a return to a society that values them.
And I think that's why I prefer DC's stories, because I view society as something as something good that we create and can make better. I'm fully aware of that African-Americans and other groups have a fundamentally different experience of their surrounding society, but to me that's reason to fix the society, not reject it as a concept.