sleeping vs slumber

sleeping and slumber both are nouns.

Word	Noun	Adjective	Verb	Adverb
sleeping	Yes	Yes	No	No
slumber	Yes	No	Yes	No

As nouns, slumber is a hypernym of sleeping; that is, slumber is a word with a broader meaning than sleeping:

sleeping: the state of being asleep
slumber: a natural and periodic state of rest during which consciousness of the world is suspended

Other hypernyms of sleeping include sleep.

sleeping (noun)	slumber (noun)
the suspension of consciousness and decrease in metabolic rate	a dormant or quiescent state
quiet and inactive restfulness	a natural and periodic state of rest during which consciousness of the world is suspended
the state of being asleep

sleeping (adjective)	slumber (adjective)
lying with head on paws as if sleeping

sleeping (verb)	slumber (verb)
	be asleep