It's not about power. Power has always been and will always be entirely arbitrary, and is not something a ruleset should look at. Groups will find the balance of power they like between the DM and the players and they'll stick to it, regardless of what's written in the rulebooks.
What is the difference between old school and new school then?
Interface.
Old-school assumes that the DM's interface with the game trumps the player's interface with the game. Sure, one player might think his character can jump the chasm since he had done similar feats before, but the player's point of view is not as important as the DM's point of view. If the DM decides that this particular 10-foot chasm isn't going to be as easy to jump as any other 10-foot chasm, then it won't be, since his view of how the chasm exists trumps the player's. What the player thought should be an easy jump is actually more difficult. This leads to a disconnect in the group, and suggests that the viewpoint of the players is unimportant.
Now this can be a good thing. The DM may not want the PC to jump the chasm because doing so would mean they would miss finding the ancient spellbook on the bridge. The game may become a million times better because the DM's point of view trumps the player's, but that is irrelevant if this disconnect disrupts gameplay enough (and in my experience, it can very easily, especially when players feel the DM is being unfair to them). It takes a good DM to adjudicate every action fairly and accurately.
New-school gaming assumes that shared interface is essential: the details of the chasm may change in each member of the group's mind (shrubs growing from the cliff-face, the color and layout of the cliff-face, the exact shape of the fisure, etc.), but the DC to jump a 10 foot chasm with a running start is always be 10. There might be a difficulty swing if the ground is unstable or there is a cross-wind, but that won't modify the DC so much that it would become impossible (+2/+4/-4/-2 modifiers being the most common). The players have the numbers to know what their characters can do, and the DM has to provide a reasonable explination as to why they cannot do so.
By codifying the way characters interact with the world, the world becomes consistent, and players become adept at gauging their experiences based on past adventures. "Woah, I rolled a 24 on my attack roll and this guy still dodged it! He must be a really high-level character! Maybe we should retreat!" translates in-game into, "Light above! This foul creature is faster than anything I've ever seen before! Pull back before it strikes again!" Players are given the ability to do what seasoned adventurers should be able to do: gauge a situation and determine whether or not they handle it. Old-school games do not offer this: when the rules change from challenge to challenge, it becomes more difficult to tell whether or not a character can successfully complete a challenge. It would be like trying to play a game of baseball where the pull of gravity changed every inning.
New-school gaming says, "Make sure the players and the DM think roughly the same thing when the word '10-foot chasm' comes up." Old-school gaming says, "Just make sure the DM knows how to handle this particular 10-foot chasm. It'd be nice of him to make sure the players are clear on it too, but don't sweat it if they don't." New-school gaming generalizes all 10-foot chasms and leaves it up to the DM to specify which one this is with modifiers. Old-school gaming specializes every 10-foot chasm, so players never know what to expect.
Outside of interface, everything else is relative. There will always be people who want rules light versus rules heavy, or storytelling versus dungeon crawling, or high magic versus low magic, or high powered versus grim and gritty, and they will exist regardless of whether the game design uses new-school or old-school interface. New-school interface will continue to be popular in groups where the DM wants to take some of the rules-burden off his shoulders while old-school interface will continue to be popular in groups where the DM likes to take the burden from the players (or the players like to force the burden onto the DM... depending on your point of view
). My group prefers new-school interface because it means that the game is consistent, making it easier on us to play the game and to interact with and gauge the game world. We dislike old-school interface because it means the DM has to be 100% clear about every challenge for us to gauge and interact with the game world.
