One notable difference between the two passages is that Lincoln believes some tension is necessary, while King calls for unity.
In the Gettysburg Address, Lincoln acknowledges the ongoing Civil War and the struggle for the nation, emphasizing that it is a test of whether the nation can endure. He frames the war as a means of dedicating the effort to the principles of liberty and equality.
In contrast, King's "Letter from Birmingham Jail" articulates the need for direct action and tension to bring about social change, while ultimately advocating for unity and reconciliation among people.