This story has a developer explaining:
https://www.polygon.com/2017/5/1/15499328/tumbleseed-hd-rumb...
The gist appears to be that the HD rumble effect allows intensity and speed to be controlled separately. Traditional rumble uses a weighted motor which shakes harder and faster as current increases. I definitely have noticed rumble effects in Mario Odyssey that feel like light, rapid trembles as well as heavy chugging thuds. There's boss fights where the sensation of the rumble actually gives you an idea of how successfully your hits are landing, and it really works. However it works, it's pretty stunning.
1) The "fizz" when you're swimming in a carbonated sea
2) Rumbling motorbikes in the city and elsewhere
3) The scrabbling feet of a "Jaxi"
These aren't the only things with rumble, just the ones that I thought felt most unique. I've also heard that in Mario Kart, the sound of picking up a coin is essentially played by the joycon's rumble, even when the audio is off.