I think part of why leveling up does not feel as massive a boost as it did in older editions is because it is not. It is due to how bounded accuracy and Proficiency bonuses usually work. I would say it does not apply as strongly to Wizards or Clerics/Druids as much as others, as they are spellcasters with tables that advance at a reasonable rate compared to older editions. This means that they are gaining considerable power when they level, as opposed to Fighters/Warriors and Rogues.
A primary example are skills. Each skill will, in theory, only increase in power every 4 levels. That leaves 3 dead levels where your skill levels are the same due to them being based upon the proficiency bonus. In older editions, they were going up (at least in 3e and after) every time you leveled up and put a skill point into them.
Another example are Warriors where each level they went up with their to hit (Attack Bonus). Between level 1 and level 2 you literally doubled in power in many instances (your hitting bonus doubled, and your HP doubled).
Except for hitpoints, this does not happen with several classes in 5e. Rather than look at increase in abilities every level, it is more of looking for a visible increase every 3-4 levels.
All classes also suffer with Saves in this same manner. The saves simply do not increase as drastically...nor do the spellcasters DC's to make it harder to save against in many cases. This also dilutes the perception of gaining a lot of power via leveling up.
The REASON I think they did this was that studies showed that the majority of gamers liked what we call the sweet spot. That spot for gaming was normally between levels 3-7. Thus, for a Attack Bonus/BAB you'd see something at maximum between +3 to +7. It is no accident that the Proficiency bonus in 5e goes between +2 to +6.
They tried to get it so the entire 20 levels are between the sweet spot. This has the effect of basically spreading out 20 levels of advancement in the same way the advancement went between levels 3 to 7 before. Thus, you have 5 levels of advancement spread over 20 levels in 5e in relation to older editions for many of the classes.