"The teacher knows the book."
Translation:Læreren kjenner boken.
Yes, they're rarely(never?) interchangeable. Most commonly:
'kjenne' is used when you know who someone by name or recognize something.
"Jeg kjenner henne" = "I know her"
"Jeg kjenner igjen stemmen" = "I recognize the voice"
'vite' is used when you're certain of something, when you have knowledge of something.
"Jeg vet ikke" = "I don't know"
"Jeg vet om en skog" = "I know about a forest"
'vet' can also be transitive: "Jeg vet det" = "I know"
'kjenner' ~ 'is familiar with'
'vet' ~ 'know', but 'vet' doesn't really work in front of a noun. You'd often (always?) use it in front of an interrogative determiner:
"Jeg vet hva du mener" or "Jeg vet hvor den ligger", not "Jeg vet meningen din" or "Jeg vet plassen".