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The Old Covenant made provision for sin to be covered, whereas the New Covenant cleanses it. It's the same difference as covering a dirty shirt with a jacket or having it totally cleansed. God does not look at us through Christ so that His blood covers our sins. Rather, Christ's blood completely removes our sin.
You're in the middle of dinner, about to go out to a special function where you are the guest speaker. You spill tomato sauce all down your front. You have two choices: you can either put on another shirt, or you can just wear a jacket to cover up the stain. The blood of bulls and goats that were such a prominent feature of the Old Covenant, could never do anything to the stain of sin except cover it up. But the blood of Jesus cleanses that stain completely.