Do narcissism and vanity have something to do with each other?

Yes: By definition, narcissistic people have highly inflated opinions of themselves and would be considered vane. However, not all vane people would be considered narcissistic.

Narcissism: The hallmark of a narcissist is an immense ego, grandiosity, and self-importance, as well as being entitled, and a tendency to use and objectify others as well as thinking of people in all-good or all-bad terms instead of shades of gray. The core of a narcissist is profound insecurity.