breathing space vs rest period

breathing space and rest period both are nouns.

Word	Noun	Adjective	Verb	Adverb
breathing space	Yes	No	No	No
rest period	Yes	No	No	No

As nouns, rest period is a hypernym of breathing space; that is, rest period is a word with a broader meaning than breathing space:

Other hypernyms of breathing space include relief, respite, rest.

breathing space (noun)	rest period (noun)
sufficient room for easy breathing or movement	a pause for relaxation
a short respite