BotW - Link is defeated, master sword is compromised, Zelda makes a big sacrifice to suppress Calamity Ganon seemingly vanishing from Hyrule, Link is awakened after after a long time gap, is provided guidance from an old spirit, needs assistance from the Sheikah to retrain his body to return to full power and is given a high tech device passed down from Zelda to aid his quest. Calamity Ganon has taken over Hyrule castle and Link must, in theory anyway, journey to the 4 regions for aid in taking down Calamity Ganon.
TotK - Link is defeated, master sword is broken, Zelda sacrifices her human form to repair the Master Sword seemingly vanishing from Hyrule. Link awakens after a shorter time gap, is provided guidance from an old spirit, needs assistance from the Zonai to remove the gloom from his body and regain his full power and is given a high tech device passed from Zelda to aid his quest. Ganondorf seemingly makes Hyrule Castle his base and lifts it into the sky and Link must, in theory of course, journey to the 4 regions for aid in taking down Ganondorf.
We start the stories at different points, the memories have a different focus, and i'd argue totk's back story is a whole lot of nothing really, and the finer details are different with a few subversions here and there but the core structure of the plot is pretty much the same.
Zelda's problem is the writing and in botw the random nature of the memories. Botw doesnt allow you to see her arc develop unless you get super lucky and they dropped the ball with the love being the key to her power shtick. In totk, she just sort of stands around in the past doing very little of anything until her transformation outside of being sad and telling people to put their faith in Link.
Zelda is a much better character in just about every other incarnation where she's actually given a spotlight.