There is no difference. The two words are synonyms. Narcissus is the botanical name for daffodils, just as ilex is for hollies.
Daffodil is the common name for all members of the genus Narcissus, and its use is recommended by the American Daffodil Society at all times other than in scientific writing.
In some parts of the country, any yellow daffodil is called a jonquil, usually incorrectly. As a rule, but not always, jonquil species and hybrids are characterized by several yellow flowers, strong scent, and rounded foliage.
But who really cares? They are all lovely flowers–and we say, “Call them whatever makes you happy!”