The word expiation begins with the prefix ex, which means "out of" or "from." Expiation means to remove something. In biblical theology it has to do with taking away or removing guilt by means of paying a ransom or offering an atonement. It means to pay the penalty for something. Thus, the act of expiation removes the problem by paying for it in some way, in order to satisfy some demand. Christ's expiation of our sin means that He paid the penalty for it and removed it from consideration against us.
On the other hand, propitiation has to do with the object of the expiation. The prefix in this case is pro, which means "for." Propitiation has to do with what brings about a change in God's attitude toward us, so that we are restored to the fellowship and favor of God. In a sense, propitiation points to God's being appeased. If I am angry because you have offended me, but you then appease me, the problem will be removed. Thus propitiation brings in the personal element and stresses that God is no longer angry with us. Propitiation is the result of expiation. The expiation is the act that results in God's changing His attitude toward us. Expiation is what Christ did on the cross. The result of Christ's act of expiation is that God is propitiated.
It is the difference between the ransom that is paid and the attitude of the One receiving the ransom.