What are the primary differences and implications of a propitiatory view of the atonement versus an expiatory view? Western theology tends to view atonement as propitiation, while Eastern tends to view it as expiation. What are the primary differences and implications of each perspective? Are they antithetical or complementary paradigms? This is related to this question, which tackles translation issues concerning biblical passages that use these terms.
I can only answer from the Western bias as I fail to comprehend the Eastern thought even when it is explained. Western thought properly includes propitiation and expiation it does not toss one out for the other or put them at competing sides.  It does not exclude either. However the Eastern thought seems to be uncomfortable with the propitiation involved in the cross and does not seem to me to believe that God’s wrath was poured out on the Son. 

In Western thought propitiation, which is that part of the work of Christ on the cross, that makes ‘appeasement’ to an offended God, satisfying his wrath for sin. The other side of Western thought is expiation. Expiation occurs on the cross (not only to have the punishment of sin removed through propitiation) but to have sins cleansed and removed as well.  Cleansing and deliverance from the actual power of sin is implied by expiation.

Cleary both of these aspects are implied by the scriptural view as the idea started with the Passover lamb and as well as the burned 'red heifer'. Both the concept of propitiation and expiation are clearly established under the Old Testament foreshadowing types. First, God’s ‘wrath’ in killing the first-born of Egypt was avoided by the Passover Lamb. The Jews were not cleansed by putting blood on the doorposts but were avoiding a deadly destruction of anger. Second, on the other hand, the burning of the 'red heifer' and collection of ashes oustide the camp clealry implies the actual ceremonial cleansing of the sacrfificial type.  The ashes of the burnt animal were directly used for ceremonial washing buy mixing some ashes with water and sprinkling the unclean. (Ref Leviticus 19, Hebrews 19:13-14))  

One can’t properly witness the ceremonial sacrifices of the Old Testament without believing both propitiation and expiation as inseparable objects, each meaningless without each other. First the worshipper or priest would lay hands on the animal to be sacrificed, symbolizing the transfer of guilt. Then the shedding of blood and total destruction of the animal would occur, symbolizing the appeasement to an offended God. Then the blood would ceremonially cleanse the worshipper. Clearly all these things support a proper view and cast grave doubts on any view that might try to limit them to only propitiation, or expiation.

Both propitiation and expiation are implied by each other in the removal of guilt. An angry God is no longer angry when guilt is transferred to another in propitiation.  The stain of guilt is wiped away from expiation. Both propitiation and expiation are meaningless terms unless the removal of guilt in understood in each.

Both propitiation and expiation are implied by each other in the removal of guilt. An angry God is no longer angry when guilt is transferred to another in propitiation.  The stain of guilt is wiped away from expiation. Both propitiation and expiation are meaningless terms unless the removal of guilt in understood in each.

The eastern view is that God changes when he ceases to be angry (according to the Western view) but propitiation does not imply any change in God whatsoever. He still is angry with sin but its guilt has been transferred to another. God can never change or remove his wrath. It must be extinguished on a a subject who bears the guilt of sin.