ball field vs playing field

ball field and playing field both are nouns.

Word	Noun	Adjective	Verb	Adverb
ball field	Yes	No	No	No
playing field	Yes	No	No	No

As nouns, playing field is a hypernym of ball field; that is, playing field is a word with a broader meaning than ball field:

Other hypernyms of ball field include athletic field, field, playing area.

ball field (noun)	playing field (noun)
the baseball playing field	a piece of land prepared for playing a game
	the circumstances under which competition occurs