Ronald C. White, who wrote a book about Lincoln's second inaugural [text here], explains in The New York Times why the speech was so great:
If his listeners expected a triumphalist address heralding a victorious North, they were instead asked to help initiate a new era of reconciliation — one marked “with malice toward none, with charity toward all.”
LINCOLN also surprised his audience by steeping his address in religious language. In those 701 words he mentioned God 14 times, referenced the Bible 4 times and emphasized the importance of prayer 3 times. The point is not to count but to listen to the way he invoked religion as a balm for a nation deeply divided.
Lincoln, so knowledgeable of the Constitution, understood that even though the founders separated church and state, America has never separated religion and politics. Lincoln is instructive in how religion can become inclusive. “Both read the same Bible, and pray to the same God,” he said, telling an audience angry at the deaths of so many sons that the South read the Bible as much as the North.
Lincoln made a final move that also set himself apart. Inaugural addresses can be exercises in self-congratulation, both of the candidate and the nation. In his second Inaugural Address Lincoln, quoting Matthew 18:7, lovingly scolded America. “Woe unto the world because of offenses,” he said. Lincoln dared to declare there was something evil at the core of this great nation: “If we shall suppose that American Slavery is one of those offenses.