The TV series itself would have been the same, regardless if Kirk was an O6 Captain or a Lt. Commander or whatever. Only the movies would see any difference, and even then, mostly just TMP, where only two years later Lt. Commander Kirk would not be Admiral Kirk. By TWOK Kirk could still easily become an Admiral, and from there on the movies could progress the same way they really did, except maybe with the Enterprise being bigger than it was on TV to justify having an O6 Captain commanding what could have gone to a Lt. Commander earlier.Not true. Again, "it is unlikely that the show would have unfolded in exactly the same way. And for the purpose of this fun thread on a fun BB, I wonder what others think about what some of those other ways might have been like."
Episodes would have been written differently and other episodes would have been written instead of some. During the series, I could see Kirk being less sure of himself and his ship (and rightly so) if it weren't the baddest ship in the fleet. I could see brinksmanship with the Klingons being more dangerous, as now it's a corvette skipper lieutenant commander dancing on the rapier's edge. I could see in episodes like "Journey to Babel" Sarek think even less of Spock's career path as he's not second in command of a "Starship," with the added significance that had in TOS, but of a perhaps less-vaunted vessel. I could see an episode written where Kirk has to go up against a full captain in command of a cruiser and show his Horatio Hornblower mettle -- a Starfleet v. Starfleet "Balance of Terror"? On and on.
But yeah, I'll echo what everyone else is saying that the main point of TOS was that Kirk was the guy in charge. He's the ship's Captain, and this would still be true regardless his actual rank. Hell, not much was different on DS9's first three seasons when Sisko was a Commander in stead of being a Captain. Even after he was promoted, aside from an extra pip on his collar, nothing substantial changed. He's still the one in charge.