The subtle distinction that makes a world of difference is that in 5e, they are only used to determine the result of a situation that has both an uncertain outcome and dramatic consequences.
This shift in thinking is why a lot of 3e (and to a lesser extent 4e players) have trouble making the shift. In 3e especially, there was an effort to take DM judgment out of the task resolution system. If you wanted to do a thing, you made a roll, because that was how the game’s engine worked. In 5e, the DM’s judgment is an essential part of the system. The player describes what they want to accomplish and how their character is trying to accomplish it, and the DM asks themselves, “can this approach succeed in achieving its goal? Can it fail to achieve its goal? Is there a cost or consequence for failing to achieve this goal?” and only if the answer to all three questions is yes does the DM call for a roll to be made. YMMV, but in my 5e games, checks are actually pretty rare, because most of the time the results of an action are easy enough to determine without that random element, and in many of the cases where the outcome is uncertain enough to merit a dice roll to resolve it, there aren’t actually any costs or consequences if it fails.
A good example of where the difference comes in is with the simple act of picking a lock. In 3e, the character comes across a locked door and says, “I want to pick the lock, can I make an Open Lock check?” and unless there’s anything that would prevent them from attempting to pick the lock like a spell or a trap, the DM would say yes. The player would make their check. On a high enough result they would pick it, on too low of a result they would fail, and in either case they wouldn’t be able to attempt again because the roll represented their best effort.
In 5e, that situation would go very differently. The player would encounter the locked door, and if they asked to roll thieves’ tools, the DM might say something like “I’m hearing you want to unlock the door by picking the lock with your thieves’ tools?” to make sure they have the correct goal and approach. Then the DM would ask themselves if the action can succeed (which it probably can, unless the DC is higher than the character can achieve on a natural 20 or they lack proficiency in Thieves’ Tools), if it can fail (which it probably can, unless the character has a particular high-level Rogue feature), and if it has a cost or consequence for failure. Now this is where things can get pretty different. If the time it takes to attempt to pick the lock is a meaningful cost (for example, if it will trigger a check for wandering monsters, or if the room is filling up with poison gas and every second counts), or if something bad happens on a failure (like a trap triggering), then the DM would call for a Dexterity check plus any relevant proficiency bonus (probably thieves tools). But if the attempt doesn’t cost the charactee anything meaningful and failure doesn’t cause anything to happen beyond the door still being locked, then it’s assumed the character just takes as much time to do it as they need and the DM narrates the action’s eventual success.
