federal department vs department of the federal government

federal department and department of the federal government both are nouns.

Word	Noun	Adjective	Verb	Adverb
federal department	Yes	No	No	No
department of the federal government	Yes	No	No	No

As nouns, federal department and department of the federal government are synonyms defined as:

federal department and department of the federal government: a department of the federal government of the United States

Other synonyms of federal department include federal office.

federal department (noun)	department of the federal government (noun)
a department of the federal government of the United States	a department of the federal government of the United States