As Marvel vs. Capcom says, "It's Mahvel, Baby!"
Here is what I see as the differences between Marvel and DC. Marvel has very down-to-Earth, sort of average Joe superheroes. If you've seen The Avengers, Thor, or heard about The Guardians of the Galaxy movie, it may seem like this isn't true. But at the heart of Marvel characters, especially all of the old school ones, are regular people dealing with extraordinary circumstances. They are easier to relate to. Spider-Man, Iron Man, Fantastic Four, Punisher, X-Men, they're all very human characters who have had some drastic change befall them and a lot of their stories are about them dealing with what's happened to them. Some books (looking at you, X-Men) have an almost Soap Opera feel.
DC has a more cosmic, world-shattering flavor. Their main characters deal with constant threats to the entire well-being of the Earth. Superman is always saving the entire world from some extraterrestrial threat. Green Lantern is really an average Joe at heart, like I mentioned above regarding Marvel, but he flies around the universe and deals with all kinds of crazy aliens. Wonder Woman is basically a god! Batman is a great character, but not as relatable as, say, Spider-Man. (Unless you're a billionaire!) I feel like the difference between Marvel and DC, in general, is small-scope vs. large-scope. DC's Vertigo line (Sandman, Hellblazer, Preacher, Swamp Thing, and many more) is a whole other entity.
I lean Marvel, but still enjoy a lot of DC stuff. Of course, they both have great characters and great stories (and they both have a lot of misses, as well). I love Batman. If you're looking for a starter book, The Long Halloween is really amazing. Year One is also very good. I would choose those to start with. If you're looking at Batman, another great series to check out is Gotham Central. It's like the old show Homicide or Law and Order, but based in Gotham. No real superheroes, just cops investigating the crimes that supervillains and wanna-bes leave behind when the superheroes are done.
Also, check out Batwoman!!! Check out the graphic novel, Batwoman: Elegy. So awesome!