Difficult to say for certain, as so much can change. Here goes nothing:
Palpatine recruits the Grand Inquisitor as his apprentice instead. He would be under a different name though - Darth (INSERT HERE). Darth (INSERT HERE) would eventually be better than Vader, as he is not riddled by medical issues or burdened by a robust, robot body.
Kanan Jarrus, Ahsoka Tano and Ezra Bridger are killed sooner than expected in Star Wars Rebels. Therefore the Rebels fail to retrieve the Y-Wing Light Bombers from Reklam Station; meaning no blueprints for the modern Y-Wing Bombers.
Senator Bail Organa now has the monumental of task of directing the insurgency by himself, but with Grand Admiral Thrawn still alive, it is stamped out fairly quickly.
Vader never travels to Tatooine, meaning Uncle Owen and Aunt Beru never die, therefore Luke has no purpose to leave Tatooine.
Ben Kenobi is hunted down and killed by Inquisitors. Han Solo and Chewbacca are still on the run from the Galactic Empire. Yoda dies on Dagobah of old age.
Princess Leia is taken by Palpatine and convinced to turn the Dark Side. She is killed when she refuses.
The Empire succeed; squashing out any insurgencies with their military might and destroying any hostile planet with the Death Star.