Shill mode engaged:
I think Song of Swords does a good job of making elves special. They've got strict conduct requirements to maintain their immortality but they're objectively stronger and faster than humans, with special powers to make them even more obviously inhuman and supernatural.
Ohanedin are cannibal nature zealots who can't touch metal or break a vow freely given. Some Ohanedin have been serving the same noble family for a thousand years on some ancient vow. An Ohanedin can put his fist straight through your chest and out the other side if he really wants to.
D&D oversaturates elves I think, which is part of the problem. They just become frilly humans. Doing things to make them less human is good, but then it becomes an RP issue if elf PCs in your game don't want to be like that.
Lord of the Rings elves, like mentioned above, are basically ancient demigods. The Mirkwood elves are scary.
Eldar in 40k, at least in the lore, have zero compassion or empathy for humans and would gladly kill a billion humans to save one elf.
The Fair Folk in the Witcher are baroque-armored immortals from another world who come here to catch slaves and hunt people for sport.
Old mythology about fairies is a good place to look usually, for inspiration like this. Badass elves should be inhuman and a little frightening. They shouldn't act the way humans do, especially when you consider the average one is a few hundred years old.
In my current 5e campaign I've removed Drow as a playable race and basically made them a cross between dark eldar and morrowind elves. Ancient fertile crescent imagery and names (The rulers of the drow empire are Marduk and Inanna), Lolth is an Old One style chaos monster instead of a regular deity, and the elves themselves are pic related. Hellraiser elves who augment themselves with fucked up surgeries and come to the surface to pillage and murder for kicks. Humans think it's an invasion, to the drow it's a safari.