The Godfather is the more critically celebrated and artistically important film.
Die Hard is a silly action film. Perhaps the best silly action film ever made, but a silly action film nonetheless.
I watched Godfather once, and greatly admired the craft of everyone involved.
I've seen Die Hard over 20 times. It's our annual "Christmas movie". I even saw it three times in a row when it was new in theaters, with my Dad.
Although you'll never heard me making the argument that it's the "better" film, it's Die Hard for me, and it's not even close.*
*[sub]I am speaking ONLY of the original film in each series.[/sub]