Alternate history:What if history had gone a different way, like the South winning the Civil War? (Or, more intriguingly: What slight change -- like a Confederate battle plan not lost before the Battle of Antietam -- would lead to the South winning?) Thereafter, how might events unfold? Example: Harry Turtledove’s How Few Remain.
Future history:A story set against a future so richly imagined that it feels like history. Examples: Isaac Asimov’s Foundation series or -- from the guy who arguably invented the concept -- Robert Heinlein’s Future History stories.