final exam vs final examination

final exam and final examination both are nouns.

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

As nouns, final exam and final examination are synonyms defined as:

final exam and final examination: an examination administered at the end of an academic term

Other synonyms of final exam include final.

final exam (noun)	final examination (noun)
an examination administered at the end of an academic term	an examination administered at the end of an academic term