playing period vs period of play

playing period and period of play both are nouns.

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

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

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

Other synonyms of playing period include play.

playing period (noun)	period of play (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