Breath of the Wild famously breaks the Zelda formula that has arguably been in place since a Link to the Past. Zelda games have generally had a pretty clear linear progression, with a formula of doing a build up of quest-like tasks that would lead you into a new dungeon, after which you would defeat a boss and then, using the new weapon or tool found in that dungeon, be able to go onto the next.
In Breath of the Wild, you start in a sort of kiddie-pool area (though combat is legitimately tougher than most Zelda games have been, so it's not like it's super-easy) where you get access to the puzzle-solving tools you'll need for the rest of the journey almost immediately.
Once you get access to the parasail, you'll be able to go pretty much anywhere in the world, and while I do feel pretty strongly that you're pushed to go to the eastern part of the map first, there's nothing stopping you from going in other directions.
There are four major dungeon-like structures, but they don't really work totally like dungeons. Combat and puzzles are somewhat more segregated - with shrines found throughout the world that serve both as teleportation nodes (along with Assassin's Creed-style map-revealing towers) and also serve as explicit "there are puzzles here" locations that award Spirit Orbs - which work mostly like Heart Pieces did in prior games (though you can also trade them in for more stamina to let you climb and run and glide longer.)
The world of Breath of the Wild is desolate. There are people - in fact, beyond the many towns you find you'll also come across little stables that serve as country inns - but the plot is all about how things basically fell apart a hundred years ago. Link - explicitly named Link in this entry, rather than that just being the default name - actually has a history, but his 100-year slumber (an emergency procedure after he took an otherwise fatal wound) has left him an amnesiac.
So you actually wind up with a few checklist items pretty early on that you can work on for the rest of the game - recovering important memories and restoring the massive Divine Beasts to bring the fight against Ganon (and give you some very useful combat powers.)
There's incredible freedom - a lot of games tout the ability to go anywhere, but in this game, with the generous (but stamina-limited) climbing, you really can get to just about anywhere you see. And also, the draw-distance more or less makes the entire game world visible (admirably without making it seem small.)
This is a game about survival, and as such a lot of the things you used to be able to count on in a Zelda game are no longer true. Simply smashing pots and cutting down grass won't get you hearts to recover your health. Instead, you'll find tons of fruits, vegetables, and meats in the world that you can cook into food. Cooking is pretty fun (you can experiment with different ingredients, though eventually you'll figure out a relatively straightforward formula) but you'll need to do a lot of it if you want to really stock up on health-replenishing items before going into a tough fight or area. As funky as the little cooking music is, this can get a little tedious.
Likewise, your weapons are not built to last. Granted, we tend to expect way more from video game weapons than their real-world counterparts (there's no way that an actual broadsword would be able to take the punishment your average WoW weapon does without breaking after like three fights,) but in Breath of the Wild, until you get the Master Sword (which itself will power-down if you use it against anything but the most obviously corrupted stuff after a while, giving it a recharge period) you'll be churning through weapons. You can strategically use weaker weapons when facing easier opponents, but just remember that nothing lasts, so don't be precious.
The open-world nature of Breath of the Wild does give you incredible freedom, but I do think you pay a bit of a price. The dungeons and shrines wind up feeling somewhat same-y. There's a lot of stuff in the open world that feels pretty unique (when you find the Lost Woods it definitely feels different) but the nonlinear nature of the progression means that a lot of the bokoblin and moblin camps kind of feel interchangeable.
There's also the fact that you have so much freedom in when you take things to Ganon that the pacing gets odd. For example, in my first play through I've found myself gravitating toward the Divine Beasts, and I left home in the middle of my fourth (the camel one in the desert.) I know there's a ton more to the game, but because everything has to be doable in any order, it means that you can burn through a lot of the "main event" stuff and then find yourself with a ton of side-missions (even if they're interesting side missions. I haven't even begun to explore Faron, for example) and a kind of nagging question of where it's time to just go and fight Ganon.
I really like the game, but I think I might have liked a more curated, guided tour of this iteration of Hyrule. The open-world aspects are arguably the fulfillment of the genre that Zelda spawned, but in embracing this freedom, the game also feels a bit more like other open-world games in ways that are perhaps not terrible, but not inspiring either.