The difference between Elf and Fairy

Elf as a noun (Norse mythology):

A luminous spirit presiding over nature and fertility and dwelling in the world of Álfheim (Elfland). Compare angel, nymph, fairy.

Elf as a noun:

Any from a race of mythical, supernatural beings resembling but seen as distinct from human beings. They are usually delicate-featured and skilled in magic or spellcrafting; sometimes depicted as clashing with dwarves, especially in modern fantasy literature.

Elf as a noun (fantasy):

Any of the magical, typically forest-guarding races bearing some similarities to the Norse álfar (through Tolkien's Eldar).

Elf as a noun:

A very diminutive person; a dwarf.

Elf as a noun (South Africa):

The bluefish, Pomatomus saltatrix.

Elf as a verb (now, _, rare):

To twist into elflocks (of hair); to mat.

Fairy as a noun (uncountable, obsolete):

The realm of faerie; enchantment, illusion.

Fairy as a noun:

A mythical being with magical powers, known in many sizes and descriptions, although often depicted in modern illustrations only as a small sprite with gauze-like wings, and revered in some modern forms of paganism.

Fairy as a noun:

An enchantress, or creature of overpowering charm.

Fairy as a noun (Northern England, US, derogatory, colloquial):

A male homosexual, especially one who is effeminate.

Fairy as a noun:

A member of two species of hummingbird in the genus Heliothryx.

Fairy as an adjective:

Like a fairy; fanciful, whimsical, delicate.