cash register vs cashbox

cash register and cashbox both are nouns.

Word	Noun	Adjective	Verb	Adverb
cash register	Yes	No	No	No
cashbox	Yes	No	No	No

As nouns, cashbox is a hypernym of cash register; that is, cashbox is a word with a broader meaning than cash register:

cash register: a cashbox with an adding machine to register transactions; used in shops to add up the bill
cashbox: a strongbox for holding cash

Other hypernyms of cash register include money box, till.

cash register (noun)	cashbox (noun)
a cashbox with an adding machine to register transactions; used in shops to add up the bill	a strongbox for holding cash