No, no. That's not what I said.
A customer can have LOTS of addresses in their address book.
They can select ONE of those as their "primary" address.
But, during checkout, on the "checkout_shipping" screen, there's an "Edit Address" button, which allows them to pick one of their address-book addresses for the shipping address.
And, on the "checkout_payment" screen, they can do the same to choose a "billing" address.
Thus, they can ship to one address, and bill to another address. (Naturally, their "billing" address really should be the one that matches their credit card address ... otherwise you'll have CC verification problems. You might want to suggest that to them.)
They just can't choose (in advance) a "default" shipping address as well as a "default" billing address ... they have to make those determinations during checkout.