The game is way less adversary. The DM's job is not to give players a hard time, but a good time. It is more about cooperation between players and DM.
Well, your millage may vary here. Any good DM would say they are running a game to give players a good time. A lot of players, if they encounter the slightest block or hardship in game play will immediately say the DM is giving them a hard time. It is really about style though: some players want to be challenged by the DM by making things hard....and some players want a more cinematic way that is not so hard. Though NS sure is all about cooperation.
It dovetails with some of the things above. For example, the reduction in lethality is not there to remove challenge, despite what many, many, many frustrating people will tell you. The point is to let you learn from your mistakes, to let you bounce back from being on the back foot, to have a difficult initial experience before rallying and ultimately winning: victory is often not ultimately in doubt, but it is initially in doubt, and that matters a lot.
The players can learn a lot from a characters death, but there is no advantage in having a character with "plot armor" survive anything. The death of a character is not the end of the game. You can keep playing.
My very basic view of the difference between new and old is that old school you'll get a puzzle to solve and players talk about how to solve whereas in new school you just roll a check to figure it out.
To restate this a bit: Old School is where the Players use their real life abilities and skills to solve problems, New School is the players use the fictional abilities of their characters to solve problems, with a roll of some dice.
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. There is a more inclusive approach to who you want to play as in 5e.
Well, not exactly. This is a bigger issue. In Old School play no one cares about the rules or what is written in the books. You want to be a goblin, dragon or whatever...go ahead.
Being a fan of the characters means understanding the characters and the situations that make them shine, and providing them with opportunities for those situations.
This is a bit too vague. And your just falling back on making everything easy and take no effort. And not really describing a "fan". Comics are a bad example too, as the hero has auto plot armor. The whole foundation of the comic is "hero has adventures each month".
There is a fine tipping point for "hard". And for most players that point is way less then most DMs, that is where the break is. I make things in a game hard....nearly impossible to the view of some players. I don't give much of an easy opening or obvious clue most of the time, but that is exactly what a fan of the characters will do.
And this again goes back to the first post about NS being Cinematic, as in most movies/tv shows the heroes are given huge obvious chances.
My New School adds that have not been mentioned:
1.New School is about shared game power and control. Each person in the game is a player, even the DM. The DM has no power to effect the character in any major way, without the player approval and consent. Players can claim everything about their character, including things such as their background destiny.
To reuse the example, in an Old School game a curse might turns a character into a duck. As DM I did not mention this to the player or in any way ask them if they wanted this event to happen. It just does. In a New School game the DM must ask the player if they want that to happen, and if the player says no, the DM must drop it. And if the player agrees they can negotiate for special effects they want in order to agree, and most often add a time limit.
2.New School is about shared creation. Everyone is free to add whatever they want to the game, without direct DM approval. And most NS DMs welcome anything the players say. You see this a lot where a group walks into a town and the DM will turn to a player and say "tell me about the local town tavern". Then the player is 100% free to make up a tavern and have it added to the game world.
3.In a New School game, the Rules are Supreme. Everyone must follow the rules at all times. If a DM wants to change or ignore a rule, it must be done with player approval. And quite often for many NS games, if page 11 in the rules says something, everyone will accept and follow it willing. There is a group feeling that very few rules should ever be changed from the "core" game rules.
And, just for the Old School side.....well, we can utterly care less what suggestions are scribbled in a book....we do whatever we want.