The main point on Lincoln's speech is that the United States must become either all free or all slave. He says that no country can survive that is divided on such a huge issue. This is both similar to and different from the ideas of secessionists.
This perspective is similar to that of the secessionists because both believe that it is necessary to have the country be all free or all slave. The secessionists wanted to break away so they could have a country that was all slave. This is not inconsistent with the idea that the country needs to be all one or the other.
The difference is in Lincoln's belief that the Union should remain united. He thought that the country, as it was then, should become all free or all slave. He did not believe that it should or would split into two different countries.