What does the bonus item from completing the Darker Side do?

After collecting 500 power moons and completing the Darker Side, you get

the Invisibility Cap. It literally makes you invisible, only leaving your shadow. Your hat, however, is visible when you throw it and looks like Mario's regular hat (with Cappy's eyes).

What exactly are benefits/effects of using this item?

This answer contains spoilers.

1. The Invisible Hat makes you invisible, except for your shadow. This has the side effect of making the game harder to play. It's more difficult to dodge attacks or land on Cappy when you can't see Mario's body. Enemies do not behave differently; they act as if they can still see you.
2. When you approach a friendly NPC when wearing The Invisible Hat, they will say "???".

Reference for #1:

Reference for #2: