In general, "billed amounts" don't mean a whole lot, since if the provider has a contract with the insurance company (usually called "in-network"), the insurance company picks what the rates are that the treatment "should" cost, and how that cost gets allocated between what the insurance company pays and what the patient pays. Due to that contract, the insurance company and provider should agree on how much the patient owes, and one shouldn't pay a provider more than what the insurance company says you should.
On the other hand, some insurance plans have a level of "out of network" coverage. In that case, the insurance company has its same sense of what the treatment "should" cost, but is really only concerned with paying its share. The EOB then should show how much they pay, but will generally show the full "billed amount" minus their payment as the amount the patient owes. That is, since there's no contract between the provider and the insurance company, the insurance company has no way to stop the provider from "balance billing" you for whatever they want beyond the amount that the insurance is paying, since any payment amount is purely a transaction between you and the provider.
In general (as with anything involving large amounts of money), it's best to get all you can in writing. The provider should be able to provide you with a statement showing zero remaining due, and including both any payment you've made to them and any payments made by the insurance company. It may take a while for the payment from the insurance company to be processed, and until it does there may be something on the statement showing "estimated amount to be paid by insurance", which isn't as comforting as one showing "actual amount paid by insurance" and "zero due".