On balance, I would have to go with Marvel, although it's actually a pretty close call.
DC only has one or two memorable characters, especially in the hero department, whereas Marvel has an entire pantheon of them. There is also a lot more variety to Marvel, or at least it seems that way due to how memorable the characters are. You've got mutants, deities, regular humans, aliens, manifestations of concepts, sentient planets...I could go on. The memorable DC characters have a great deal more depth to them than a lot of Marvel characters, but they never actually do something with them. DC comics bore me, and I have tried - so many times - to get into them.
Even with the New 52 reboot a few years back DC is still dull, dull, dull. It's gritty and far more realistic, but that's all I can really say in its favour. DC is the sort of thing you can almost believe happening in real life sometimes - especially in the case of Batman - and, to me, that just makes it far less engaging than Marvel's sheer ridiculousness. Marvel Now! was a far better reboot; it streamlined everything and made the crossovers that much better.
Marvel comics are more dynamic. They've got better writing, better art (for the most part) and far more intricate plots, which occasionally cross over for a pretty epic collision which rocks the entire universe. Marvel actually does things with its characters, which is why its comics are so much better. The multiverse (because hey, why just one universe?) is constantly changing, and the fallout from events like House of M and AXIS turn the entire thing upside down. EVERYONE is affected in some way by the major plot arcs, and I love this. You get maybe one or two issues that tie-in to another story going on elsewhere, and then things go back to normal, only with that little aftershock from the other event. Everything feels so much more connected, and characters react to events that they don't even have a big part in. DC, by comparison, has always felt very isolated with its comics, and with a few exceptions, the writing is quite stale...and the characters just aren't as compelling. I love the one-offs, like Superman: Red Son, and occasionally things like The Killing Joke will shake up things, but...well, these are one-offs. Marvel is doing this constantly, across multiple comics, simultaneously.
In terms of movies, Marvel wins this as well, hands-down. I really didn't care for Nolan's Batman movies and, whilst the earlier DC movies were pretty good for their time (no, seriously) the MCU is a juggernaut DC are going to be hard-pressed to beat if the best they can come up with is Batman vs. Superman. The MCU plays out like the comics - it even draws inspiration from the better storylines from them - and, whilst it treats its villains horribly, so far it's been a pretty thrilling ride, with more promised. DC, by comparison, has been far too slow in both revealing its planned movies and releasing them. It hasn't got a hope in hell of catching up, much less matching, at least in my opinion.
That said, DC have infinitely better cartoons, and they always have. As good as the 90s Spiderman and X-Men cartoons were, the interlinked DC cartoons of that era - Batman, Superman, Static Shock and Justice League - were far, FAR better, and Batman: The Brave and the Bold was a superb show, giving DC some much needed comedy. By comparison, the current era Spiderman, Avengers and Hulk (ugh!) cartoons leave a lot to be desired. Marvel haven't had a decent cartoon made in years and, whilst the recent Batman cartoon was a flop, I eagerly await the next one they'll come out with. DC know how to make cartoons.
In terms of TV series, I'd say the two were about equal. Agents of SHIELD is great, but so is Gotham, and Smallville was fantastic. Both Marvel and DC are doing great jobs with their TV series, and are promising great things to come (I'm perhaps a little more excited for DC than Marvel, but only because Krypton has so much potential, and the Daredevil movie put me right off the character so I just can't bring myself to get excited for it) and at the moment, for me, it's too close to call to say which I prefer. It depends on what they come out with, and how well SHIELD sets the stage for the Inhumans movie, really.
Comics and movies, Marvel. Cartoons, DC. TV shows, about equal. Marvel wins for me.