From the information available, what exactly changed during the 2nd Rebellion where the rebels won, was that by the time the rebels invaded the Capitol, the Capitol's military had already been decimated. With the bombing of the Nut (the Capitol's military headquarters) in District 2, the Capitol's control over Panem was pretty much over at this point (except for the city itself, of course!). In the film, following the bombing of the Nut, resulting in the Capitol losing what President Snow described as their "best troops", Snow himself acknowledged that the game was over at that point, and that they were toasting "a glorious era, coming to its bitter end". At that point, Snow began making preparations to face the rebels in the invasion of the Capitol, ordering his officials to ready anti-aircraft defenses, and the Gamemakers set up pods to cause large numbers of deaths to the rebel forces.
So, in conclusion, the turning point for the Second Rebellion was when the rebels launched airstrikes on the mountains surrounding the Nut in D2, triggering avalanches, burying the military base, and having nearly completely destroyed the Capitol's military (the air force of the Capitol was done, by this point). THIS is what sealed the Capitol's fate, and led to the war turning in favor of the rebellion.
By contrast, during the First Rebellion, it is mentioned that the rebels attempted to scale the Rocky Mountains to invade the Capitol, but were easily spotted and attacked by the Capitol's air force. According to the Hunger Games Wikia (and Katniss in the first book), this loss was a major factor in the First Rebellion's failure. As the rebels sent large numbers of soldiers to climb the Rockies to attack the Capitol, and losing large numbers in the attempt, this lead to the downfall of the First Rebellion. So apparently, when the rebels tried to invade the Capitol in the first rebellion, they had NOT completely destroyed the Capitol's military (it seemed like the Capitol was still quite strong then, compared to the 2nd rebellion, when the rebels completely destroyed the Capitol's military through the bombing of the Nut). I think this was probably the difference in the first and second rebellion.
I wonder, though, since during the First Rebellion, D13 was the military-industrial complex of the Capitol (D2 was the military-industrial complex in the second rebellion), and D13 had pried from the Capitol's rule, WHY was the Capitol still able to win, and use their air force to destroy the rebels trying to climb the Rockies? If the Capitol had gotten all their military equipment from D13 at the time of the First Rebellion and D13 was siding with the rebels, why was the Capitol still the winner at the end?
Maybe the Capitol had other military bases too, elsewhere throughout Panem?