While I adore Die Hard and Die Hard with a Vengeance, and love them more than any Fast and Furious flick, FF is just the more consistent for me. There are five Die Hard flicks and I only truly like two of them. There are moments in the other three that i do like, but that series takes a nosedive anytime someone not name John McTiernan directs one. Also doesn't help that they've turned McClane into a superhero.
On the flip side, turning the FF crew into superheroes has done wonders for the series. While Fate of the Furious is meh for me and sorely misses Paul Walker and some energy from anyone in front of the camera (besides Charlize), that series has been on a role since part 4. From 09-16, Justin Lin and James Wan dropped nothing but heat, with Fast Five being one of the best action flicks this decade. Fast and Furious has shown the ability to reinvent itself, something Die Hard sorely needs.
I'll watch 2 Fast, Fast and Furious, Fast Five, Furious 6, and Furious 7 before I watch any Die Hard that's not 1 or 3.
Much like the Alien vs M:I matchup, this one is about consistency of a franchise vs great movies. Die Hard 1 and 3 take dookies on any fast and furious flick, buuuuuuut FF has more good movies on its resume and that counts a lot for me.