Also, how would you say 1st Edition is different than 3rd? I don't have much experience with D&D in the TSR days. How is class balance, and how much is character power tied to wealth and magic items? How detailed is the combat system?
I think people have failed to mention what I consider the core difference: the underlying party dynamic. First edition is built around character interdependency. A cleric can't do what the fighter does. He can FIGHT, but there's no way for him to buff up and become a better fighter than the fighter himself. The magic-user has spells, and those spells are quite powerful, but he's extremely vulnerable in a fight--MUCH moreso than in third edition. The thief (no P.C. terms like "rogue," thank you very much!) has value, because for the most part, other classes don't have quick-and-easy ways to replace his core competencies. (The magic-user still has spells like Knock, but in a game where magic-users don't get bonus spells, he's not going to be flinging Knock left and right.)
It's still all about resource management, but there's a lot less margin for error. Magic-users get no bonus spells at all for high intelligence; clerics get a handful of bonus spells for high wisdom, but nothing like what they get in third edition. EVERYONE gets far fewer hit points; you only have hit dice up to level 9 or so, and then a flat +1, +2, or +3 per level after that. Only fighters can get a Con bonus of higher than +2. A huge, ancient red dragon only has 88 hit points, but that's enough that you do NOT want to mess with him. (Especially since his breath weapon does as many hit points as he has left.)
Certain monsters are considerably scarier. Level-draining undead do just that: you get hit, you lose a level. No "temporary negative levels." Poison tends to kill rather than inconveniencing.
This isn't to say there's not room for optimization. Multi-class characters can be quite powerful; they advance in both (or all three) of their classes simultaneously, and divide their experience points between them. The upswing is that, while most of the party is 9th level, you might well be a level 7/8 fighter/thief. You're not quite as good at either class as the single-classed characters, but you're a lot more versatile.
Dual-class characters are about the closest you can get to 3.5 multiclassing, and they're uncommon. Only humans can dual-class, and they have to have at least a 15 in the prime stat of their first class and a 17 in the prime stat of the second class. (Keep in mind, stat boosts of any kind are extremely rare.) You start as one class, then switch to a second class, and when your second class level exceeds your first, you get to freely use the abilities of both classes.
AD&D isn't smooth. It isn't polished. There are a lot of things that seem kludged together. However, I actually think it's better-suited to long-term play than third edition. A lot of the "illogical" things they changed in third edition make sense, from a balance perspective. For instance, every class has a different experience chart. A thief advances far faster than a magic-user. That's not particularly sleek or streamlined, but it DOES lead to more balanced characters, as the 10th level thief can more easily hold his own with the 8th level wizard.
I'm actually a big fan of some of the OSR games that have said, "Let's take what was good about first edition and figure out how to fix the illogical parts." Castles and Crusades is good; OSRIC is also good.
But I'd gladly dust off my first edition rulebooks and play a new campaign.