at the same time vs all the same

at the same time and all the same both are adverbs.

Word	Noun	Adjective	Verb	Adverb
at the same time	No	No	No	Yes
all the same	No	No	No	Yes

As adverbs, at the same time and all the same are synonyms defined as:

at the same time and all the same: despite anything to the contrary (usually preceding a concession)

Other synonyms of at the same time include even so, however, nevertheless, nonetheless, notwithstanding, still, withal, yet.

at the same time (adverb)	all the same (adverb)
at the same instant	despite anything to the contrary (usually preceding a concession)
overlapping in duration
despite anything to the contrary (usually preceding a concession)