final exam vs examination

final exam and examination both are nouns.

Word	Noun	Adjective	Verb	Adverb
final exam	Yes	No	No	No
examination	Yes	No	No	No

As nouns, examination is a hypernym of final exam; that is, examination is a word with a broader meaning than final exam:

Other hypernyms of final exam include exam, test.

final exam (noun)	examination (noun)
an examination administered at the end of an academic term	the act of examining something closely (as for mistakes)
	the act of giving students or candidates a test (as by questions) to determine what they know or have learned
	a detailed inspection of your conscience (as done daily by Jesuits)
	a set of questions or exercises evaluating skill or knowledge
	formal systematic questioning