I only got the "this is next gen" feeling twice in my life, and both of them were relatively close to eachother. One was Dead Rising on the xbox 360. It was so densely packed with enemies and interactive objects, a clock that governed everything, and tons cool little things absolutely everywhere. Many games during that generation I thought could have their fidelity scaled down, and they'd still work pretty much the same. As the Chop Til You Drop port proved however, once you scale this game back, you end up with something very different, and a lot worse.
The other game for me was Assassin's Creed. The way everything moved, and Altaïr found his paths throughout the massive game world was so impressive to me. You just pointed him into a direction, and he would just try to go, regardless of the obstacles in his way. They refined it a lot in the sequels, but I remember being so immensely impressed at how far this technology advanced compared to the Prince of Persias that came before. This type of pathfinding made it possible to integrate your possible routes so much more seamlessly into the environment than before. Heck, many games after it still don't do it as well as the first Assassin's Creed. I see games all the time where you can where designers clearly carved out a route, and it often is your one and only approach. Breath of the Wild's climb anywhere philosophy isn't as elegant looking at this, but it certainly scratches a similar itch. It feels very liberating to not have your routes spelled out, because you can rely on the traversal mechanics being consistent throughout the entire game. None of this "you can climb this thing now, but not this other identical thing later" noise here.