If "Yesterday's Enterprise" is to serve as a guide, all of Federation (and regional) history would have changed. It's impossible to know exactly what the results would be, but we can speculate based on what the absence of Kirk, the Enterprise, and her crew, might mean if they were left out of key moments in history.
Consider how the following events might've occurred without the Enterprise:
"The Doomsday Machine" - without Decker's suicide-by-shuttlecraft accidentally revealing a way to deactivate the alien machine, it could have devasted the Rigel system and beyond.
"The Changeling" - without Spock mind-melding with Nomad, and Kirk persuading Nomad to self-destruct, this probe might have been even more destructive than Decker's Planet Killer.
"The Immunity Syndrome" - without the good ship Enterprise being in the neighborhood to investigate the fate of Gamma 7A and the Starship Intrepid, the rogue space amoeba may have had time to gather energy and multiply.
TMP - When V'ger came calling, what would Earth have done without the refit Enterprise and Spock to mind-meld with V'ger? (aka "Where Nomad Has Gone Before".)
TVH - Assuming Earth did survive all that, how would it have fared against the Whalesong Probe without Kirk and his merry band of renegades stealing a Klingon scoutship?
TUC - Without Kirk on the scene, would the Enterprise-B (or whatever it would be called) have been able to survive the Nexus and rescue Guinan?
So, without Kirk and company and their mighty starship, the Federation and the Galaxy might've been in deep doo-doo.