How is the word "self-conceited" different from "narcissistic"?

According to vocabulary.com, the word "Conceited" is defined as:

A conceited person has an inflated self-image and perceives himself as incredibly entertaining and wonderful. Talk incessantly about your accomplishments on the clarinet or amazing ability to wiggle your ears, and people are going to think you’re conceited.

It pretty much sounds like "narcissistic". However, I sense that there could be a subtle difference of connotation, or nuance, of those words. Could anyone tell me how those words are used in real life?

Solution 1:

Narcissistic comes from a morality tale about vanity so I'd use it for someone too vain for their own good.

'Conceited' has more of a sense of arrogance and unjustified self-regard.