I know a big part of the difference in damage opinion was due to season 5's tank meta. When half the team is tanks, and games would commonly go 45 minutes to an hour long damage seems a lot lower, not because damage is actually lower, but because tanks were more effective. Tanks didn't die as quickly, and the more tanks you have on your team the longer it takes to kill said tanks, leading to very long drawn out fights.
Mobility was also lower, meaning that if you got away from fights you could actually escape more often, leading to fights not being one sided stomp fests and resulting in having people alive to deter pushers more frequently.
While damage is not much higher, the speed at which things die is significantly higher, and your chances of being able to escape and reset when behind is lower than before. Snowballing is also has a much higher impact than previously.
Towers also die far more quickly now, for a multitude of reasons.
Overall the game has sped up, fights are shorter, people die more quickly and have less chance of escaping to fight back. This isn't because damage is significantly higher than before (although it is, that was around the time Riot changed from their nerf OP to buff weak strategy) but because durability is lower, mobility is higher, and snowballing is far more extreme.