If you purchase a BIN, that is through Ontario only and you can put it on your receipts instead of your SIN to make your clients happy. It only means that you own your business but does not give you any copyright over your name.

The Business Number is Federal and is only required for people who have a payroll, so employees working for them. They have a whole other set of rules. That's the basic difference, but you can do an internet search and look at it for yourself easily. Again, your accountant is wrong.