From what I see by looking at the two words "katallaso" and "apokatallaso" is that the former speaks to a conciliation (as mentioned by others), and the latter speaks to salvation by the conjoining of God and man by the Spirit, creating The Body.
1 Cor 7
10 But to the married I give instructions, not I, but the Lord, that the wife should not leave her husband 11 (but if she does leave, she must remain unmarried, or else be conciliated (katalasso) to her husband, and that the husband should not divorce his wife.
The conciliation of the woman to the husband is her part. The husband's part is to "not divorce his wife" (i.e. remain conciliated to HER). This reminds me of how God was in Christ conciliating the world to himself and gave us the ministry of reconciliation: "be ye reconciled to God."
Here is apokatallaso and salvation by way of being reconciled-together by Spirit:
13 But now in Christ Jesus you who formerly were far off have been brought near by the blood of Christ. 14 For He Himself is our peace, who made both groups (Jews and Gentiles( into one and broke down the barrier of the dividing wall, 15 by abolishing in His flesh the enmity, which is the Law of commandments contained in ordinances, so that in Himself He might make the two into one new man, thus establishing peace, 16 and might reconcile (apokatallaso - end the separation) them both in one body to God through the cross, by it having put to death the enmity.
17 AND HE CAME AND PREACHED PEACE TO YOU WHO WERE FAR AWAY, AND PEACE TO THOSE WHO WERE NEAR; 18 for through Him we both have our access in one Spirit to the Father. 19 So then you are no longer strangers and aliens, but you are fellow citizens with the saints, and are of God's household, 20 having been built on the foundation of the apostles and prophets, Christ Jesus Himself being the corner stone, 21 in whom the whole building, being fitted together, is growing into a holy temple in the Lord, 22 in whom you also are being built together into a dwelling of God in the Spirit.
So, the conciliation is the changed nature of God's relationship to man (not that God changed, for he never changes), being that he has put away the enmity, which was the law of commandments contained in ordinances requiring Jews and Gentiles to be distinct by way of genetic traits and physical circumcision.
The reconciliation (apokatallaso) is the fruit of the conciliation, the actual conjoining of God, Jew and Gentile into one new man by the Spirit into the body of Christ through faith.
10 For if while we were enemies we were conciliated (katalasso) to God through the death of His Son, much more, having been conciliated, we shall be saved by His life.
That last part is the apokatallaso - saved by his life, through conjoining of the Spirit into the body, thereby giving us access to grace saving us from sin. The word "apo" in the Bible is usually translated into "departed" as in when Jesus "departed (apo)" from one part of the land to another.
Apokatallaso solves the problem of people being departed from God, where katalasso solves the problem of not having a way to conjoin the two.