At various times England was invaded by Romans, Anglo Saxons, Vikings and then the Normans. Each left their mark on English vocab which means often in English you have different words which express the same meaning because English vocab imports words from German, French, Latin etc.
"Invoice" is derived from the French word
envoi = to send. Envoy comes from the same root.
"Bill" probably comes from the Latin
bulla = a document.
You could also use "account" which came to us via the Latin
computus meaning a calculation, which became à compter, then
accompterin French. Ditto "Facture" which comes from the French "Facture" or German "Faktur"
At then end of the day it all means the same ......you have to pay up