What I find most interesting is how 80s it feels and how gory it is. Die Hard is the father of the modern action movie -- everyman hero instead of shirtless Stallone (although Bruce Willis spends much of the movie in a tank top or shirtless while carrying a machine gun) and with a lot of blood squibs exploding in people's pants. But although it's the father of the modern action movie, it's very much also a movie of its time.
It's neat seeing the early work in a new creative movement, before it completely rejects the tropes of the old. It's interesting noticing all the things post-Die Hard action movies reject that Die Hard itself didn't.