In your context "billing" is to make an itemized list or statement of fees or charges. "Invoicing" is to actually present (send) that list the customer.
I may run up a MasterCard bill, for example, but I do not pay until I get the invoice.

I wanted to add that most dictionaries use the terms interchangeably, though. In the business world, however, some companies use both terms to indicate that their billing system is separate from their invoicing system (usually SAP).