Part of the problem with animal sacrifices is that a consciousness of sins remains. The Israelites went through the ceremony, but spiritually, nothing happened. Christ's sacrifice, though, includes the removal of sins from consciousness, which is implied in the role of the azazel but not actually accomplished. Along these lines, the author of Hebrews notes three times that Christ's sacrifice cleanses the conscience (Hebrews 9:9, 14; 10:22).

Hebrews 10:4 uses significant wording when speaking of taking away sins. The Greek word, aphaireo, has familiar meanings and implications. Its basic meaning is “to remove,” while it can also be translated as “cut off.” The author uses terminology that exactly describes the azazel, the goat of “complete removal” (as The Brown-Driver-Briggs Hebrew and English Lexicon renders it) that bears the sins to a land that is “cut off.”

Certainly, nothing involving animals can do that. Christ's sacrifice, though, was both necessary and entirely sufficient to deal with sin. He became sin for us (II Corinthians 5:21) and was cut off. He paid the death penalty, giving us access to the Father, as well as bearing the sin into forgetfulness and cleansing the consciences of those who accept Him.