to a higher place vs higher up

to a higher place and higher up both are adverbs.

Word	Noun	Adjective	Verb	Adverb
to a higher place	No	No	No	Yes
higher up	No	No	No	Yes

As adverbs, to a higher place and higher up are synonyms defined as:

Other synonyms of to a higher place include above, in a higher place.

to a higher place (adverb)	higher up (adverb)
in or to a place that is higher	in or to a place that is higher