period of play vs playing period

period of play and playing period both are nouns.

Word	Noun	Adjective	Verb	Adverb
period of play	Yes	No	No	No
playing period	Yes	No	No	No

As nouns, period of play and playing period are synonyms defined as:

period of play and playing period: (in games or plays or other performances) the time during which play proceeds

Other synonyms of period of play include play.

period of play (noun)	playing period (noun)
(in games or plays or other performances) the time during which play proceeds	(in games or plays or other performances) the time during which play proceeds