What is the difference between an elf and a gnome?

It depends upon your frame of reference.
If you're talking traditional folklore, then elves are esseentially forest spirits, immortal and fae.
https://en.wikipedia.org/wiki/Elf
Gnomes are shorter humanoids, and earth dwellers.
https://en.wikipedia.org/wiki/Gnome
In Tolkien, gnomes are a type of elf, the Noldor, or deep elves.
https://en.wikipedia.org/wiki/Noldor (it's at the bottom)
In D&D they are distinct races, elves associated with forests, and gnomes with burrows (and magic). However, there are so many sub-races of each that all you can really say is that elves are taller and live longer.
In modern culture, well, Santa's elves and garden gnomes? Who can tell the difference?