Nintendo Land was great for the asymmetrical multiplayer. Four players co-op on the big screen versus a fifth player on the GamePad. It's not impossible to do in other ways, but I haven't seen it done.
I think shared/split screen on the TV adds an additional social element to it. Otherwise it's just all of us playing on their phones/Switch.
Mario Maker used the touchscreen effectively. You could seamlessly go back and forth between editing your level on the touchscreen and testing it on your TV. I also liked the touchscreen effects in Super Mario 3D World and regret their (effective) absence in the Switch port.
Wonderful 101 does it beautifully, entire game was designed to use it in many different ways.
The best overall use is the Nintendo DS emulator. Unfortunately only few games are available for purchase but with a bit of hacking the compatibility is great.
Yes, really innovative use of the gamepad for ZombiU gave the game a properly scary aspect when checking in bags and boxes, and nervously looking up at the screen as you're doing it.
In Tokyo Mirage Session you get text messages from the characters on the second screen so it's like you are glancing at your cell phone as you play. This worked very poorly when they ported it to Switch and you had to go into a menu when a party member texts you.
I felt like later in the Wii U's life, they were pushing it more as being for off-TV play, and that meant that that kind of stuff would have to be implemented anyway— Wind Waker HD certainly had the dual approach, where the default was to play on the TV with inventory on the gamepad, but if you played in off-TV mode, then the inventory was a pause menu popup. That extra development work to support both was probably invisible to most users who only ever played on-TV.
That combined with the increasing inevitability that everything was going to be ported to a later, single-screen system anyway, and it's not hard to see why titles like Donkey Kong Tropical Freeze and Zelda BOTW didn't use the secondary screen at all.
Splatoon— second player without split screen.
New SMB U— gamepad player can spawn blocks to help you reach secrets.
Wind Waker— map and inventory.
But an awful lot of titles just mirrored the TV or had a button to enable off-TV play.