Points of Light
The 4E Dungeon Master's Guide defines the parameters of a typical D&D campaign in the setting section. The default setting is called Points of Light, which describes how the world is mostly wilderness, full of monsters and ancient ruins, and peppered with occasional safe havens (typically villages and small cities).
The 4E game is exception based. Where AD&D defines rules for this and rules for that and a different class might have its own system (e.g., thieves), the core 4E rules are very simple. On top of that very simple core, add exception after exception. If you've played Magic: The Gathering, you've seen this principle in action.
To great effect, 4E defines classes primarily with powers. Every class works pretty similarly in this regard. The difference is in the list of combat powers that defines each class. Each power provides an exception to one or more rules, to be used in specific combat situations.
Most versions of D&D have some idea of non-combat skills that any character can learn. In AD&D 1st Edition, it's the "secondary skills" table in the DMG. In 2nd Edition, it's non-weapon proficiencies. 3E and 4E add a true skill system. The game calls for you to roll these skills all the time. 4E adds a skill challenge system for handling non-combat obstacles that require effort over an extended duration.
I and many other people love the idea of skill challenges but hate the 4E implementation. Find Stalker0's Obsidian Skill Challenge system and use it instead. It's fantastic and solves all my problems with the built-in system.
To Level 30, Tiers
AD&D didn't have level limits, but it got pretty unwieldy after a certain point (some say 6th level). By unwieldy, I mean that the DM's effort to create encounters just gets harder and harder as the characters level up. 4E makes encounter building a lot easier.
4E also takes the characters to level 30 with clear sets of powers for each level. It does this in three, 10-level tiers: heroic (1-10), paragon (11-20), and epic (21-30). Play changes a bit at each tier, and the kinds of encounters PCs face (and where they adventure) changes considerably.
AD&D combats mostly turn into characters in fixed positions trading blows with monsters until one side loses. There's not a lot else going on. In 4E, the standard/move/minor action system gives players a lot of options every turn. A typical round for a rogue includes moving around (avoiding opportunity attacks by shifting), attacking, and maybe some kind of minor action to boot. A lot happens in six seconds! (Oh yeah, combat rounds are six seconds, not a minute.)
4E is way more gonzo. Or laser-sharked. Whatever term you use to describe it, 4E has lots of weird races and classes as PC options. Play a goliath ardent, a dragonborn sorcerer, a tiefling warden, etc. The old races and classes are still there; don't worry. Just expect spikes and horns on everything (not really, but it sure feels that way).
4E introduced the idea of distinct power sources to the game. AD&D differentiated between types of magic, and dumped illusionist magic in with magic-user magic but separated them from clerical divine magic. 4E gets explicit about where a character's power comes from, even if he's a fighter. You have the martial power source for fighters and rogues, the arcane power source for the magic-user types, divine magic for the clerics and other religious types, primal power for druids and the like, and the psionic power source for a host of mentally-driven characters.
Defenses, Saves, Death
All different! Be careful here.
You have four defenses: Armor Class, Reflex, Fortitude, and Will. Basically they took effects that used to require a saving throw to avoid and turned it around. Now you have a defense against those things. The fireball attacks your Reflex. The illusion attacks your Will. You don't have to roll anything to avoid it. They either hit or not.
And while we're talking about AC, gone are AC to-hit charts and THAC0. You roll d20 + bonuses and try to meet or exceed the defense (AC or otherwise) to hit. Simple enough.
Saves still exist, but they're for shaking off lasting effects. Say you're taking ongoing damage from being on fire. At the end of your turn, you get to try to save against the effect. All saves are the same: roll d20 and get a 10 or higher. Simple as pie. Mmmm, pie.
If you fall under 0 hit points, you're unconscious and dying. Every turn in combat, make a saving throw. 10+ and you don't get worse; 1-9, and you get one strike. Three strikes, and you're dead. Roll a 20, and you get to use a healing surge! Oh, the negative-10-and-you're-dead thing changed. Now you get to go negative equal to your "bloodied" value, which is half your maximum hit point total. Pretty sweet.
In general, 1st level characters are pretty damned tough. However, the monsters they face are pretty tough, too. Your 1st level rogue might have 23 hit points, but the kobolds have 27. (Just making up numbers, but you get the idea.)
Encounters in AD&D (and really every version before 4E) were pretty much focused on one small area, typically a single room. You go into the 10x20 room and fight monsters. 4E encounters sprawl over multiple rooms because of all the extra movement. It really feels like a big fight, dragging in monsters who overhear the noise. Or you might loop around the back way to flank the boss.