Traditionally, in a Zelda game, you smash pots or cut grass to replenish your health and supplies. Zelda never had a rebounding health meter the way some games did, but healing up didn't take a lot of effort, and getting things like arrows after you ran out was not the most difficult task.
In Breath of the Wild, you need to conserve like crazy. Even the important weapons that you get from beating major bosses will break after a bit of use. Link is constantly picking up new stuff, and if you go into a dungeon, you're going to want to have supplies - not just arrows and a full arsenal of weapons, but also good food to heal up with.
Cooking takes time, and sometimes you find that all the standard ingredients - apples, mushrooms, etc. - are out and all you have is the makings for some kind of buff elixir - which are fine, but not what you want to see when you're looking at a very small number of hearts left.
Breath of the Wild is not linear. I've been following what seems like the natural path - I went to the Zora and helped them get the Elephant Divine Beast online again and then headed north to Death Mountain to help out the Gorons and their Salamander thing. For all I know, the game steers you in this direction, or it's possible that I could have just gone west after Kakariko Village and done entirely different things.
The Ancient Beasts serve, I assume, as Breath of the Wild's main dungeons. These massive constructs (the open world of BotW recalls Shadow of the Colossus, and these beasts certainly seem of a similar style to the Colossi) must first be neutralized, which seems to be a unique quest chain for each, and then you go inside and solve puzzles.
While there are little bits of corruption to fight inside the Beasts, dungeons in Breath of the Wild are really puzzle-focused. Fighting is for the outdoors, and you'll get plenty of combat against Bokoblins and Moblins (and other things, though it's mostly, at least so far, those guys) as you explore.
While the Divine Beasts are almost entirely puzzle-based, they each end with a boss fight. Strategies are actually a bit less obvious in this game, given that you have all the "tool" abilities pretty much from the moment you leave the starting area.
Restoring a Divine Beast to order will not only bring its aid against Ganon, but you'll also get a passive effect based on the champion who piloted it a century ago. These seem like pretty powerful bonuses, which really incentivizes you to get those beasts, even though technically I think you can actually go straight to Ganon pretty early in the game.
I do have a little worry that I'm focusing too much on the main quest. In open-world games (and to an extent Zelda games have always been open-world games. You could argue they were the first major example of the genre) I am often driven to see the plot and not get overwhelmed by the many, many side activities one can do. With two major bosses down after having the game for only a day and a half, I do want to stop and smell the roses a bit. I guess the question is which roses first.