final examination vs exam

final examination and exam both are nouns.

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

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

final examination: an examination administered at the end of an academic term
exam: a set of questions or exercises evaluating skill or knowledge

Other hypernyms of final examination include examination, test.

final examination (noun)	exam (noun)
an examination administered at the end of an academic term	a set of questions or exercises evaluating skill or knowledge