I would never make a claim that one is better then the other, because I do not believe that one is actually better. My preferences tend to lean toward Marvel but I do have many beloved DC characters that I will always want to keep up with and read about. Both companies are diversified in the stories they tell as well as how they tell them. What, to me, is the fundamental difference between them is that Marvels characters are children of science and DC's are children of drama. When comparing the origin of characters most of Marvels heroes and villains received their powers through scientific mishap. For DC most have origins mired in deep personal tragedy or conflict.
Let us examine DC's characters 1st.
To discuss DC means that the big two have to be touched upon. Both Superman and Batman are characters defined by loss. They are both orphans whose missions are defined b y their personal tragedy. Looking beyond them though, you will see drama weaved through out the DC universe. Wonder Woman was created from clay because Hippolyta wanted a child. Green Arrow was stranded on a desert island. Green Lantern (every Green Lantern) receives their ring because the Lantern before them died. The Martian Man hunter is similar to Superman in that he to is the last of his race. Tragedy is the pervasive them affecting the heroes as well as villains and even sidekicks.
DC villains like Lex Luthor, The joker and Sinestro all have terrible tragic pasts filled with jealousy, betrayal, wounded pride and shame. Of course what makes them a villain is that they look inward and seek to make society pay for the wrongs they have received. As far as sidekicks, I wouldn't want to be on to a DC hero. Every Robin has had their parents die either before or after their 1st meeting with Batman. Green Arrows former sidekick Speedy was a heroin addict and his current sidekick is an H.I.V. infected former teen prostitute. So lets face it things are pretty dramatic in the DC universe.
The Marvel universe as we know it is predominantly due to one man. Every comic book fan knows to who I am referring, Stan Lee. All the major characters that stand a top Mount marvel are his brain children, for whatever reason Stan Lee founded all his characters in some sort of scientific background. As it was the height of the cold War, nuclear proliferation and the space race in retrospect it makes sense why so many marvel heroes were dosed with radiation, beginning with The Fantastic 4, Lee's 1st major success which began the birth of the modern Marvel universe.
As most know Reed Richards, Sue and Johnny Storm and Ben Grimm were exposed to cosmic radiation, resulting in their powers. Next came The Hulk, Bruce banner the survivor of a gamma bomb explosion. Of course the pinnacle of the irradiated hero is Spider-man, who was bitten by a radioactive spider.
Not all heroes in Marvel need to be exposed to radiation to be imbued with special powers. Tony Stark built his Iron Man armor to escape captivity and save his failing heart. Captain America 9not a Stan Lee creation) is the product of a Super Soldier Serum. What might be on of Stan Lee's greatest ides was that of evolution/mutation in creating The X-Men. He admittedly did this so he could create new heroes without having to constantly come up with a unique back story to explain their powers.
The difference between Marvel and Dc can be summed up as such: Marvel characters and stories are based in science fiction with great does of drama. DC on the other hand is based in drama with doses of science fiction.