DC is better
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The DC universe has darker, more challenging storylines
DC storylines are darker and edgier than Marvel, making them much more compelling. Darker storylines are more challenging to the viewer because often, the characters face uncomfortable truths and realities which challenge their views and ideals. This, in turn, challenges the audience's own views and ideas.
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The Argument
DC's stories carry with them a certain darkness - a willingness to explore tough philosophical questions, uncomfortable societal topics, and even the most shameful parts of human nature. These types of stories are much more intellectually and emotionally stimulating than an average Saturday morning cartoon.
Take the famous Injustice series, for example. It follows how Superman, the most powerful hero of the Justice League and a paragon of virtue and justice, develops into a dystopian tyrant. After the Joker kills Lois Lane, Superman decides that he needs to take justice into his own hands, and use his power more punitively on evil. But this mindset warps him into a villain, and as members of the Justice League take sides with or against him, their union dissolves into a violent schism that rocks the whole world. This storyline explores themes of corruption in a unique way by exposing even its most beloved heroes to the same temptations that cripple so many leaders in the real world.
Another example is the famous Christopher Nolan film, The Dark Knight. Its antagonist, the Joker, is driven by the belief that all people, despite their outwardly moral appearance, are just one bad day away from devolving into ruthless animals. He attempts to prove this by plunging the city of Gotham into chaotic violence, forcing everyday citizens, law enforcement officials, and even Batman himself, to make extremely difficult choices and sacrifices that challenge their very notions of right and wrong. The movie shows human nature at both its best and worst, and its ending is far from a "happily ever after." Instead, it challenges the viewer to grapple with the Joker's claims and confront the fear of evil in real life.
Compare these stories with Marvel's quippy comedy and generic save-the-world plots. While those stories do have their place, when it comes to quality, DC has them outclassed.
Counter arguments
Marvel has its own share of dark storylines. Very similar to DC's Injustice is the Civil War arc, which follows the clash between the members of the Avengers over how much power superheroes should have over the world. While Iron Man professes that governmental authority should be paramount, Captain America rebels against the authoritarian aspects of Iron Man's rhetoric. Heroes take sides and clash in a bitter conflict.
The stories of the X-Men are filled with conflict which challenges the viewer to think about social issues. In the world of the X-Men, humans fear those among them called mutants, born with genetic enhancements to their bodies and abilities. As a result, mutants find themselves under oppression from the government and society. While some mutants, like those under the leadership of Professor X, try to forge peace, others under the influence of Magneto decide to strike back in retaliation. These stories are filled with examinations of prejudice, acceptance, identity, and community. They depict fear, hatred, and ignorance in a very real and relatable way.
Those are just some examples of Marvel's dark storylines. Furthermore, "dark" need not always be better. A story can have a lighthearted tone and still teach valuable lessons. Marvel excels at this; while their movies are known for quippy humor and common plots, they still manage to teach valuable lessons through Iron Man overcoming trauma, Thor learning responsibility, and Star Lord discovering his family, along with many more examples.