The advantages of hiding rolls is that the master can create suspense and drama. When the players are exposed to the mechanics, they can start making decisions that affect their characters in ways that their character wouldn't act if the player didn't have that information available. The advantage of letting players roll is that it makes it easier for a master to focus on the story since players can help take care of some of the mechanics; the game generally moves faster when the players are helping. Also, players generally like to feel in control, so letting them literally decide the outcome by way of rolling is a pretty natural extension of granting that feeling of control (commonly called "player agency").
For games that hide some rolls, the purpose is to avoid leaking too much information. For example, if a player rolls a 1 on Perception, they might start utilizing other skills to confirm if it is really safe, even though the general definition of a 1 would be that the character is blissfully unaware and unlikely to perform any further checks for their safety. Some people even consider this to be cheating, because they're using information they shouldn't have. Responsible role-players might be able to act out their character by separating character knowledge from player knowledge, but casual players probably won't, and take advantage of the information intentionally or subconsciously.
So, while skills are different from game to game, rolls should generally be hidden when it would (a) expose information to the players that would change the character's behavior, despite the character not having that knowledge, and (b) it would heighten the sense of drama when something unexpected happens. You can always choose to expose a roll afterwards, but you can't "hide" it afterwards. If a player is incredulous about a roll, you can still choose to present the evidence if you want to (unless it's a fudge, which masters should reserve only for dire consequences without a deus ex machina save, preferably no more than once or so a game).