Both of these movies are two of my personal favorites really.
The American Film Institute ranked Die Hard as the #1 action movie of all time and I can see why really. During the 80s, jacked up guys like Schwartzenegger & Stallone were everywhere in action movies. They played these damn near superhuman, indestructible badasses that could take on a hundred men without breaking a sweat. Bruce Willis in Die Hard was someone that as a badass but he was a realistic badass. He didn't look like he spent 12 hours a day in the gym. He couldn't face down a couple dozen men armed to the teeth simultaneously. However, Willis was a different action star here than his contemporaries. Willis' character was a tough, determined, courageous and resourceful New York City police officer that found himself in a situation that was way over his head. Whereas Arnold & Sly's characters seemingly were in such situations, they rose to the occasion with comparatively little trouble. Willis was someone that endured fatigue, nerves and was injured at various points during his jaunt through the Nakatomi building. Willis' character was witty as well and the story of Die Hard is classic action movie plot but the execution and the characters involved aren't stereotypical action movie stuff and that's a big reason of why it works and stands out.
Sin City is this gritty, stylish thriller that had something of a Pulp Fiction type of feel for me with the way the storylines all connected together during the movie. The look of the movie was something that was different and what I really enjoyed most was the dialogue of the characters. Characters using word and phrases like "dames", "broads", "bum ticker" seem like they belong back in the 30s but I like it. The characters are really what make the movie for me, especially Mickey Rourke's portrayal of Marv. Even though Rourke is older and doesn't quite fit the hulking description of Marv, I can't imagine anybody else playing it. His gravelly voice during Marv's inner monologue just fits the character perfectly. This movie is also the first time I remember seeing Jessica Alba and I have to admit that my heart stopped just for a bit. Damn, she was the finest thing I'd ever seen.
I'd say Die Hard will win here because it's really become an action classic over nearly a quarter of a century.