There's indeed a difference between 3.x and 5 paradigm
After a little research I came to conclusion that the role of rules in 3.5e and 5e does differ:
3.5e rules serve like a rigid framework, that explicitly says, what mechanics we must use and when
5e rules are more like a toolset, where choosing the right tool is a
subject of the DM's discretion
How Skill checks are supposed to be made in 3.x
3.5 edition goes from the concept "character uses a skill":
When your character uses a skill, you make a skill check to see how well he or she does.
Pathfinder define this even stricter, explicitly describing when "you must make a skill check":
When your character uses a skill, he isn’t guaranteed success. In order to determine success, whenever you attempt to use a skill, you must make a skill check.
So, the player say "I use my Search skill", for example. Then you have to roll what the game rules want you to roll. Then you get a particular result from the list pre-written in the rules. Then you use this skill again (and roll once more) because the circumstances changed. This could lead to a lot of unneeded dice rolls.
This is not how skills work in 5th edition
The latest edition removes "skill checks" completely. It was not just about renaming "skill check" to "abilty check with a proficiency bonus", it's actually a new approach (an old one, actually — that's how the rules were used in 2.x edition) — the DM asks for an ability check (or uses other dice mechanics) when they think it is necessary:
When the outcome is uncertain, the dice determine the results.
See these questions for more details:
The "How to play" chapter describes the game process in three steps and does not even mention any mechanics. Instead, players are supposed to describe, what their character do in the game world. DMG says the game can be played without dice at all (the only exception is combat), if the DM thinks there was no uncertainty to determine, as being said in "The Role of Dice":
The DM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure. When the outcome is uncertain, the dice determine the results.
The rules use words "might" or "can" in most cases regarding checks which are made out of combat:
the DM might call for a Wisdom (Animal Handling) check
5th edition empowers the DM in ways that 3rd, 3.5, and 4th did not. While rule zero has always applied, 5th edition chooses not to explicitly codify many things. DMG address this in the beginning of the first chapter:
The rules serves you, not vice versa
Jeremy Crawford, the lead game designers, supports this idea in several interviews, (for instance, this Dragon Talk around 45:15).
"...we tried to make every piece of the game to feel like a tool that players and DMs can pick up and use with consistency"