The South would probably lose more quickly, it might only take four or five years instead of the six of OTL.
Well, all that was necessary was to bleed the North white and eventually Washington would change its mind. One has to question Lincoln's resolve in appointing Grant as general-in-chief since all he did was throw his soldiers against Thomas' defences and the Union Army thru the meatgrinder.
While the Confederacy did lose its claims to the New Mexico Territory and West Virginia independence was achieved, tho that was more thru Hooker's coup de'tate in Washington and Stanton's short lived 'Committee of Public Safety'. What is deplorable, before the European's intervened to force the peace, where the Government ordered executions, at Stanton's order, of at least 120 civilians when martial law was declared in New York, Boston, Philadephia, Columbus and Chicago.
Granted the 'long truce' between the Union and the Confederacy was shakey for the first four decades, but eventually things settled down and there was strong commercial ties between the two nations by 1900. Both Americas sent forces overseas to aid the Allies during the Great War and both fought alongside each other in the Great Pacific War against Japan and Imperial China.