【Expiate】 , 【atone】  mean to make amends or give satisfaction for an offense, a sin, a crime, or a wrong.

The same distinctions in implications and connotations are observable in their derivative nouns expiation and atonement.

【Expiate】  and expiation imply an attempt to undo the wrong one has done by suffering a penalty, by doing penance, or by making reparation or redress.

【Atone】  and atonement have been greatly colored in their meanings by theological controversies. The basic implication of reconciliation became mixed with and sometimes subordinated to other implications (as appeasement, propitiation, or reparation). In general use  【atone】  (usually with for) and atonement emphasize a restoration through some compensation of a balance that has been lost.

When the reference is to an offense, sin, or crime the words usually imply expiation, but they stress the rendering of satisfaction for the evil that has been done by acts that are good or meritorious; thus, one expiates a sin by doing penance for it, but one atones for it by leading a good life afterwards.

Sometimes a deficiency or a default rather than an offense may be atoned for (as by an excess of something else that is equally desirable).