This leaves the United States, north and south, in a state of legal limbo. Neither side is willing to back down, but the status quo doesn't work for either side, as the southern states at this point are in rebellion (although not active), but not their own nation yet. Neither side has yet acted as an aggressor. We probably see a cold war which eventually leads to active violence when things boil over.
Hard to see who benefits from this war. If I remember my chronology correctly, this prevents the south from alienating northern radicals, but also prevents states like Virginia from siding with them.
edit: this goes in rpe-1900, by the way