computer vs computing machine

computer and computing machine both are nouns.

Word	Noun	Adjective	Verb	Adverb
computer	Yes	No	No	No
computing machine	Yes	No	No	No

As nouns, computer and computing machine are synonyms defined as:

computer and computing machine: a machine for performing calculations automatically

computer (noun)	computing machine (noun)
a machine for performing calculations automatically	a machine for performing calculations automatically
an expert at calculation (or at operating calculating machines)