From the get-go, Horizon hits you up front with a supremely well-told introduction to both Aloy and father, Rost, alongside some basic backstory on the neighbouring Nora tribe and the world at large. Overall, Horizon's tale is one of mystery - you're discovering why you don't have a mother, what this weird, futuristic underground 'Metal World' is, and just how we arrived in this strange post-apocalypse of leather-bound warriors and high-tech dinosaurs.
In Zelda, story is absolutely minimal. You awake in an underground tomb awash with high-contrast blue and orange hues, pick up your inventory and ability-enabling Sheikah Slate, have a few words with a mysterious Old Man who tasks you with saving Hyrule, and set off to do so.
Along the way you'll pick up more information as to what happened, who you were and how you got there, but the overarching narrative is old-school 'unassuming warrior tackles unspeakable evil'. The best story moments of Breath of the Wild are dynamic, coming from your own interactions with its endlessly experimental world, not from the occasionally flat cutscenes or interactions you have to see.
To that end, when all's said and done in both games, you'll still be emotionally drained, but Horizon asks some incredibly profound questions as to who we are as a species and how we think, before giving you some incredibly well-researched exposition dumps that fill out every last lingering question you might have.
Winner: Horizon Zero Dawn