I've played a bit of Destiny. I'm somewhere around level 24, still too low for the big ol' Vault of Glass, and I noticed something. Nothing changes after the... I want to say Cabal are revealed. The big space marine lookin' folks with the shields. Every fight is roughly the same grouping of enemies. Two or three groups of three to five small enemies, with up to three bigger enemies. Bigger fights have a few more groups. Repeat as needed. Then there's the bosses. The bosses, as far as I can tell, have all just been stronger versions of other units in a special arena. Much stronger. Dozens and dozens of whole clips of ammunition. Near-instant deaths. There's a number of people who call this sort of difficulty 'fake' because it doesn't offer anything new.
Duration and failure speed are how some games do difficulty now, I guess. It just requires you to do the same thing over and over, introducing opportunities for minor slip-ups to cascade out of control. Increasing the duration of a task (Every boss's health in Destiny and every 'keep this car alive while driving it half way across the city' mission in GTAV,) just introduces more chances for the player to fail, while raising the failure speed (High damage guns, weak car,) shrinks the envelope of safety and makes every mistake more dangerous.
It can be done well, and there's no exact formula for doing it well, but players shouldn't have Repetitive Strain Injury by the time they've finished an encounter. It's best used to draw out a task long enough to highlight the next several points I'm going to bring up.
So how do we make things hard, then? I ask, rhetorically, as I pretend to be you. You make the player do something more mechanically complex than the regular game. Make targets smaller and move faster, put some bright red weak spots in weird places, or give your enemies strange new weapons for the player to dodge. This serves the purpose of keeping your player engaged, as every one of these harder encounters will require that much more effort.
New weapons, however, brings us to the next point. At some point, your awesome, difficulty-increasing event will become rote for the player as they internalize how to deal with its flavor of challenge. This is why novelty is important. A player only has the tools you give them and the behaviors you teach them, so something new will be, at the very least, a momentary challenge while they figure out how to deal with it. This also means that you can't just constantly throw the same challenges at your player. After the initial few encounters with their new friend, the player needs to have time away from it. Spreading the introduction and usage of your mechanical challenges is key to keeping your player from perfecting their ability to mow through everything you throw at them.
Lastly, variety. Mix your challenges up a bit. Every new combination of old challenges is, itself, a new challenge. That's the easiest thing.