in public vs publicly

in public and publicly both are adverbs.

Word	Noun	Adjective	Verb	Adverb
in public	No	No	No	Yes
publicly	No	No	No	Yes

As adverbs, in public and publicly are synonyms defined as:

in public and publicly: in a manner accessible to or observable by the public; openly

Other synonyms of in public include publically.

in public (adverb)	publicly (adverb)
in a manner accessible to or observable by the public; openly	in a manner accessible to or observable by the public; openly
	by the public or the people generally