The Gauntlet: https://www.youtube.com/watch?v=cOtOnFqibZ4
On Prestige Mode, there's a few changes. First, each time the runner has to change. When a person completes running through the Gauntlet, they are given a debuff called "Proven Worthy" which requires another player to run.
Second, in addition to the regular shielded Psions spawning when the person in the Gauntlet passes through a wall, a Psionic Projection will also spawn in a random location near that plate. Both need to be killed.
Lastly, in the final running phase when everyone has to go through the Gauntlet, there's only 3 orbs on each wall instead of 4.
Pleasure Gardens / Dogs: https://www.youtube.com/watch?v=j4zUmoyJX80
The dogs now all have new pathing which they take, and instead of there being 6 dogs in total there are now 8. The 2 new dogs appear at L2 and Cave when you're going to damage them. Otherwise the rest is the same.
Bathing Ritual / Royal Pools: https://www.youtube.com/watch?v=MWhxNp-Re9c
The main difficulty in this fight is that the enemies are significantly harder to kill, and also do a hell of a lot more damage. Mechanically, the only other changes are: The Bathers now drop a purple pool on the ground upon death that will lower your Psionic Protection if you stand in it. And after you do a damage phase on all the lamps and have to do another cycle, 4 players are then given a debuff named "Burden of Worthiness".
To remove this debuff, you need to look plate in the middle of the room for the icon that appears (Dog / Chalice / Axes / Spears (?)). You then have to go to the corresponding plate for the icon you see, and that allows you to pick up the Psionic Protection and continue the fight. If you do not do this, you will die to the debuff.
Emperor Calus: https://www.youtube.com/watch?v=m5-0d0WkenQ
There's only one difference on this fight but it does make things a lot more chaotic. Whenever a Psion is killed in the Throne room (the shielded / symbol Psions), the person who killed it will be instantly teleported into the Shadow realm. A random player from the Shadow Realm will then be teleported back into the Throne room in their place. Otherwise, the enemies just hit harder, and Calus has slightly more health. We managed to one phase him on this, so his health isn't a big difference - but the random teleports definitely makes things a lot more chaotic.