Gamma World is a stripped-down subset of D&D 4e with a few tweaks and a few new add-ons. I grabbed the basic rulebook and came up with this list. There are more options in the two expansions, mainly Secret Societies and Vocations, and you can find other fan-generated house rules online.
Setting: It's post-apocalyptic, not fantasy. Anything goes. The Big Mistake has mashed all alternate realities into one broken mess of a world, so you can find anything you dream up somewhere out there in the wreckage. People made of rock, sentient rat swarms, and psychic powers are all commonplace here on Gamma Terra.
Character Generation: Randomly roll for 2 origins which fill the function of race and class. Mash them together however you see fit. You get primary stats, powers, skills, and special bonuses from each. Fill in an 18 and a 16 for your primary stats or a 20 if they're the same stat, then roll 3d6 for each non-primary stat. You also get training in an additional random skill.
Characters can progress from Level 1 to Level 10, not 30 as in 4e. There are no tiers, Paragon Paths, nor Epic Destinies.
Powers: There are no Daily powers in Gamma World. Every power is either At-Will or Encounter. Every origin has Novice, Utility, and Expert powers. You start with the Novice powers from both of your origins, giving you 2 At-Will powers. (EDIT: Some Novice powers are Encounter powers, but they're usually At-Will.) You gain other powers (almost always Encounter powers) as you level up: Utility powers at levels 3 and 7; Expert powers at levels 5 and 9. You get to choose which origin's power you take at the lower level, then get the other one at the higher level. Each origin also has a Critical benefit which give extra damage or effects on a critical hit. You earn Critical benefits at levels 2 and 6. Both effects trigger on a critical hit after you earn the second one at level 6.
The Cards: Gamma World has 2 types of cards which gives your character extra Encounter powers during play: Alpha Mutations and Omega Tech. They're presented as cards because they change so often and they're randomly awarded. The rules suggest that each player make their own Mutation and Tech decks to draw from during play, but using GM-controlled decks work just fine. Both Mutations and Tech grant your character additional Encounter powers, but one's intrinsic and the other is an item of some sort.
You start with one Alpha Mutation card at level 1, but get an additional card at levels 4 and 8. Whenever you roll a 1 on any roll you discard one Mutation card and draw a new one. After every encounter, you discard all of your Mutation cards and draw new ones. You can also Overcharge powers to increase their effectiveness, but you need to roll a save: failing the roll earns you a penalty instead.
Omega Tech cards are the equivalent of magic items. They persist, but after you use an item you roll a save to see if it resets for the next encounter or if it burns out. Once an Omega Tech item burns out, you can salvage it to get a fairly powerful piece of regular equipment. Salvage items have a level rating - if your character isn't at least that level, you can't salvage it and it's discarded.
Skills: There are only 10 skills. They're broader than 4e skills and invite GM and Player interpretation. For instance, Science covers all knowledge from meteorology to particle physics to geology to first aid checks to stabilize dying characters.
Combat: The combat rules are stripped-down 4e and covered in about 12 pages. You still have surprise, initiative, d20+attack mods vs. Fort/Ref/Will/AC defenses, and Standard/Move/Minor actions from 4e. The rules cover basic attacks, opportunity attacks, combat advantage, flanking, cover, concealment, move actions, forced movement, difficult terrain, and conditions. The rules don't cover charging, disarming, tripping, or grappling, so there's plenty of room for a GM to improvise or add rules straight from 4e.
There are new damage types as well, most notably Psychic and Radiation. Certain origins give resistances to certain damage types, but it seems like all the damage types are so common in the environment that you'll never be protected enough.
Hit Points and Rest: Gamma World still has a Bloodied score and when you go negative by that much, you're dead. It still has a death save. It does not have healing surges: you get a single Second Wind per encounter which restores your Bloodied score in hit points. A short rest restores all hit points and powers, so there's no reason to opt for an extended rest except it's required before leveling up.
Weapons and Armor: Gamma World gives you very broad categories of weapons and armor, and it lets you fill in the details. Weapons come in 6 categories: Heavy and Light varieties of Melee Weapon (baseball bat, lamp post, sharpened street sign), Ranged Weapon (rock, repeating crossbow, sharpened rebar), and Gun (derringer, minigun, plasma rifle). Melee weapons give slightly more damage if you use them 2-handed. Guns do more damage than Ranged Weapons, but require ammo which is very abstract. If you use a gun once in an encounter, you're conserving ammo and have some left. If you use it more than once, you're out at the end of the encounter and can't use your gun until you find or buy more.
Armor comes in Heavy and Light varieties, plus Shield. Light gives +3 AC and lets you add your Int or Dex bonus to AC, Heavy gives +7 AC but lowers your move by 1, and a Shield gives +1 AC and restricts you to one-handed weapons.
Monsters: The monster stat blocks in Gamma World are almost exactly the same as 4e, and I've heard of people using 4e monsters in Gamma World with no modifications at all.
Attitude: The rules encourage you to go over the top whenever possible. It's a light-hearted but nihilistic game at heart - "...your character dies. Too bad, so sad." Character creation is so fast that it dares you to kill off the PCs because they'll roll up new characters in the time it takes the rest of the party to finish the encounter. And did I mention you're standing in a radioactive pool right now? Take ongoing 10 radiation damage, save ends.
It's a minimalist system which invites interpretation and improvisation from GMs and players alike. Compared to 4e, Gamma World gives you a sketch of a character's capabilities and dares you to try to break things.