In 1864 Lincoln apparently considered replacing Hamlin with a War Democrat in order to get more votes, partially out of the fear pre-Sherman's March that the North was sick and tired of the war and wanted out. Among the various considerations were Andrew Johnson (a native of Tennessee and Military Governor there) and Benjamin Butler (New Orleans military governor). Assuming that Lincolns assassination goes off as OTL, what would either of these men handle the post war situation? Obviously Booth and Co. would be hanged, but would either of these men try anything like that on the Southern Generals or the more rabid Fire Eaters?
I think the better question is what does this do regarding the "Klan Wars" of the 1870s. Denounced in their time as "Carpetbaggers" and "Scalawags", the transformation of former slaves into equal citizens of the United States came at a very high price--but if the only way the Union could be held was through the strength of free Coloreds, so be it.
Men like Hamlin and his successor, Salmon Chase, aren't going to move from "brigands who murdered the President and deliberately attacked us" to "people we can make a deal with". And I'm sure that Butler, who had quite the record in the Klan Wars, wouldn't have shied away from that challenge.
But Johnson? The man is a southerner and largely didn't share the Lincoln-Hamlin-Chase mentality of uplifting the former slaves and ensuring a fair deal for them. The guy was an opposition figure by 1866, for goodness sakes. I think if he were president, the Confederates would get a peace that leaves them in control and neoslavery would remain intact for generations.
The Butterflies could be catastrophic from this alone; there obviously wouldn't be President Booker T. Washington elected in 1904; heck, without President Washington, fundamental social services enjoyed by most of the Western World probably wouldn't have been passed for a full generation--instead, racist hick bastards would simply divide and conquer, isolate the blacks and use them as a wedge to prevent basic social needs from being implemented.
Can you imagine how many Triangle Shirtwaist Fires there would be if Washington wasn't able to use the outrage to make fire safety codes a national requirement? How many laborers would be gunned down if the right to organize and build unions wasn't elevated to a cabinet office and defended by heroic republicans?
I think we've done rather well. Thank goodness for no Johnson--he would have been bad and might have been truly awful.