This is a long post with lots and lots of words and no pictures at all.
Yesterday I described newer RPGs as "loose." That was unfortunate because I did not define my terms closely and I received thousands of cards and letters here at the HQ from confused, frightened and angry young people from all of the six corners of the Realm of Men. (The Realm is hex shaped, of course!)
Well children (under adult supervision while you use The Computer, of course), let us talk about what is loose in the old school and the new school and what is rigid.
In the old school, resource management was paramount: time, torches, weight, distances, money, hit points, and so forth. Especially calendar time, if you meant to make a campaign out of it. But at least time measured in Turns. Everything could be related to encumbrance, time, and XP and they were all closely related to one another (see this post here.) Even leveling up, which is instantaneous now in the time of the ubiquitous computer RPG, took some money and some weeks training.
In the new school, time, distance, and encumbrance are almost-universally hand waived away. I think that's because the math is tedious and managing resources gets in the way of telling a fancy story. Money is no longer even nominally tied to XP. Fighting monsters is a central activity rather than one of last resort.
So in this respect, old school play is "rigid" and new school is "loose."
In the old school, there are very few numbers and words on the character sheet. Players can't rely on tools the rules have given to him in order to overcome puzzles and challenges. Therefore the player has to think up the solutions. Overcoming mysteries, traps and puzzles is more like a parlor game of 20 Questions than a mechanical exercise. In fact, "metagame knowledge" can sometimes be allowed or even encouraged in this kind of game, in the name of rewarding clever players. Old school proponents will call this "playing face-up" because most of the time you are working things through with your fellow players and the ref rather than looking down at your sheet.
Combat is described loosely. Weapons are simple; combat "maneuvers" such as they are, are just telling the ref what you want your guy to do. Maybe they get as complicated as splintering your shield or setting your spear.
In the new school, character sheets are jam-packed with lots and lots of detailed information covering a lot of corner cases. Many, many puzzles and mysteries are overcome with dice throws rather than worked through like a parlor game. We sometimes call this "playing face-down" because your nose ends up buried in your character sheet!
This is especially true for conbat. Weapons and tactics are defined very precisely in new school games. This is because the authors expect combat to be the central activity. "Builds" - the way you construct your character beforehand - become more important than an agile mind at the table (not that an agile mind is unnecessary!)
So in this case, old school play is "loose" and new school play is "rigid" - especially when it comes to combat!
One of the knock-on effects of this emphasis on individual builds and intricate skill lists is that rolling up a character takes a lot more time for the newer games. For 0D&D or Swords & Wizardry, you can put your man on an index card in just a few minutes. For a polished 3.5E character, you might spend four hours or more getting him exactly right!
While there's nothing wrong with complicated applied math problems (I love making superheroes up in the HERO system), it means a new guy takes way too long to roll up at the table. And that means that everyone there is quite a bit more hesitant to see a guy die. And that means the real possibility of character death becomes much, much smaller in the new school. And if that fear is far removed, the stakes go way down and potentially so does the fun.
That is a pretty fundamental difference: the rules of the older games focused on tracking resources. So the challenge was to extract XPs from the dungeon (in the form of wealth, mostly) before one of your critical resources ran out and you had to/got to roll up a new guy.
The challenge of the new school is to construct a combat machine to participate in a series tactical skirmish minigames, and to enjoy being a content tourist in someone else's world. By the way, that's not a bad thing. Most video game RPGs are built this way, and millions of people love to play them!
Note that both old and new schools allow for plenty of character development and interaction. I think that this is its own reward, even though some systems additionally reward it with XPs. But I think we should save exploring the philosophy behind PC and NPC interactions for another day.