telegrapher vs telegraph operator

telegrapher and telegraph operator both are nouns.

Word	Noun	Adjective	Verb	Adverb
telegrapher	Yes	No	No	No
telegraph operator	Yes	No	No	No

As nouns, telegrapher and telegraph operator are synonyms defined as:

telegrapher and telegraph operator: someone who transmits messages by telegraph

Other synonyms of telegrapher include telegraphist.

telegrapher (noun)	telegraph operator (noun)
someone who transmits messages by telegraph	someone who transmits messages by telegraph