Both of these words refer to the idea of coming first, but in different ways.
The adjective premier means the first, highest status, most important, or best:
Jimi Hendrix was the premier guitar player of his time.
As a noun, premier can mean “prime minister” for certain nations.
Premiere can be an adjective or verb referring to the first performance or showing of a work of art, or of an artist:
For her premiere show, the director decided to premiere her new play.