The difference is that a bill is saying "I've done this work for you: now pay me". Or "I've given you these products: now pay me". But an expense claim is sayi

The difference is that a bill is saying "I've done this work for you: now pay me". Or "I've given you these products: now pay me". But an expense claim is saying "When I was doing what you asked me to do, I had to pay some money on your behalf, so please refund me".