Best Hero
Marvel: Spider-Man

Peter Parker, a brilliant scientist in addition to being a pretty OK photographer, designed a pair of wrist-mounted web shooters which produce a strong, sticky web-like substance, allowing him to sling webs all over New York City. It is these web shooters that make him truly recognizable as "Spider-Man," rather than "Sticky Bendy Guy." Spider-Man is haunted by Personal Demons, which force him to fight crime to atone for allowing a thief to escape a crime scene and eventually kill his dear Uncle Ben.
To sum up:
Powers: Spider agility, spider strength, spider sense, wall-crawling, implausible inventiveness, guilt.
Weaknesses: Rolled-up newspapers, redheads
DC: Superman

As you can imagine, it's hard to build suspenseful stories based around a hero who has seemingly infinite power and only one weakness. There are only so many "villain with a chunk of Kryptonite" stories that can be told. Sure, Doomsday "killed" Superman, but Supes only considered that to be a temporary setback. He and his new Super Mullet were back in action quicker than you can say "Christ-like figure."
Superman's human identity is Clark Kent, mild-mannered reporter (superheroes like working at newspapers, because that's the quickest way to get news. Or at least it was, 40 years ago. I suppose nowadays it'd be more efficient to monitor the blogs. But honestly, I don't think anyone wants to read a comic where the hero hangs out with Perez Hilton while waiting for Amy Winehouse to start lighting kittens on fire). Kent works at The Daily Planet, along with his on-again off-again girlfriend/wife Lois Lane. His childhood sweetheart was Lana Lang. His nemesis is Lex Luthor. There must've been a heck of a deal on the "L" blocks for typeset printers back in the '30s. Superman's main problems include feeling sad about not being able to save everybody all the time, and having too many women fall in love with him at once (more on that later).
So here's the breakdown on Superman:
Powers: Invulnerability, super strength, super speed, flight, X-ray vision, heat vision, super cold breath, easy-to-print primary color scheme
Weaknesses: Kryptonite, accident-prone reporters, Jesus complex
If you're asking who's more powerful, it's Superman hands down. But if you're asking who's more of a hero, it's got to be Spider-Man because he's actually risking injury when he fights evil.
Winner: DC
Worst Heroes
Marvel: Daredevil

Powers: Blindness
Weaknesses: Pretty much anything he would have to see to avoid. Bullets, knives, baseball bats, trash cans, steel chairs, angry cats, cars, bikes, trains, joggers, etc.
DC: Aquaman

Powers: Talks to fish (not that they listen), can swim almost as fast as Michael Phelps.
Weaknesses: Air, loneliness, freedom, network mergers
I've got to give the edge to Marvel for having the slightly less useless character. Somehow, despite having superpowers, Aquaman manages to be less effective than a blind man. He's the John Tesh to Daredevil's Ray Charles.
Winner: Marvel
Best Team
Marvel: The Uncanny X-Men

DC: The Justice League of America

I'll give this one to the X-Men, because they use more of an actual team mentality. The Justice League is basically just Superman and some other people. If the JLA were a band, Superman would leave and put out a solo album.
Winner: Marvel
Other Media
Marvel: Marvel has released many more movies over the last decade than DC, but many of those were clunkers. Yes, the Spider-Man movies were great (Spider-Man 2 is arguably the best comic book movie ever), but that doesn't excuse Ghost Rider, Daredevil, Elektra, Hulk, X3, or the atrocious Fantastic Four movies. Iron Man earned Marvel a lot of the goodwill it lost with its many bad films, and it looks like the company is trying to interconnect its films in the future (I'm already excited for the Avengers movie, and Iron Man 2 hasn't even started filming). As far as TV shows, Marvel has faired much better with cartoons than live-action shows. Many people of a certain age fondly remember the X-Men and Spider-Man cartoons of the '90s. Marvel has yet to create a viable live-action TV show based on their characters. It's time to step up.
DC: DC hasn't had much luck with its superhero movies, and since they only use a few of their characters in movies the bad ones really stick out. We had to deal with 10 years of terrible Batman movies (yes, even the Tim Burton ones). Finally, though, Christopher Nolan has gotten the franchise where it should be, with Batman Begins and the upcoming sequel, The Dark Knight. Fanboys have been arguing about Batman Begins vs. Spider-Man 2 as the Best Ever for a few years, and The Dark Knight will likely be added to that debate. The Superman movies, well...oh boy. The original Superman movie would have been great if not for the ending, involving Superman flying backwards around the Earth, reversing Earth's rotation, and therefore somehow reversing time, which apparently is tied to the rotation of the Earth. Try not to think about it too hard, or your brain will bleed. Superman 2 was OK (enough so that I aped one of its more famous quotes for my blog title), except for the part where the President of the United States giving General Zod control over THE ENTIRE PLANET. I'm sure the U.N. was THRILLED about that move. The less said about the other Superman sequels, the better. Superman Returns could've been great, but since Brian Singer doesn't actually READ Superman comics, he based the movie on a pre-Crisis framework. To translate for you non-nerds, that means that he made a movie based on the 1970s Superman lore, rather than the very-different canon of the new millennium. It'd be like if Batman Begins had been based on the 1960s Batman series. Bad move. Not many other DC characters have had their own movies, so the company lives and dies by Batman and Superman. How many times do I have to ask for a Green Lantern movie? RotoScope it. It'll be awesome. TV-wise, Smallville has been going strong for seven seasons now. For those of you who haven't seen it, Smallville follows the problems of a young Clark Kent. Most of his "problems" revolve around the premise that pretty much every woman he knows is madly in love with him. Yeah, I know. I feel sorry for him, too. It's just more proof that Superman has never had a real problem in his life.
It's tough to call this one, because of the variety of media and the differing amounts of output between the two companies. DC, though, has produced quality results in movies, live-action TV, and animation, whereas Marvel has more mixed results in movies and has had virtually no success with live-action TV. I'll hand this one to DC.
Winner: DC
So it looks like it's all tied up. I guess I'll have to break the tie here. Guess what folks? Gorilla Grod is NOT a Marvel character. He's straight outta DC, and since I'm aping his name I'm giving his creators the edge.
DC FTW.