Well, the systems are really in different paradigms, but I'll boil some of the differences down for you.
First the concepts you have to throw away.
Attributes. There aren't any. There is no way to say that character a has a strength of 18 and character b has a strength of 15.
Levels. Again, there aren't any. There is a way to judge power level, but it is not so rigid as a d20 game.
GM Fiat. Fate is a collaborative game more than d20. The GM is not always right; the players of the game (including the GM) come together to make an interesting narrative. That's a good segue to the last bit.
d20 is in most cases rules first. You can play it other ways, but as written, it's bent towards the rules being the arbiter, with the GM as the final arbiter. Fate is narrative driven. Your use of the rules is driven by the narrative, rather than the other way around.
Direct equivalencies do (sort of) exist
Skills. Skills are one of the members of a trifecta that describe a character. They tell what a character is good at- pretty similarly to d20. Even in that, they help define your character in a way... your skills are defined in a pyramid, with one skill being your peak skill. So your warrior with mighty thews isn't likely to have Lore as his peak skill unless there's a good reason behind it.
Feats = Stunts. Well, as I said, sort of. Stunts are special traits that change the way that a skill works for you. Normally, you couldn't use your fight skill to dodge an arrow... but maybe you trained specifically for that in your background, and have a stunt to reflect that. Or maybe you're better with one type of weapon over others... stunts do that, and a whole lot more. They form the second part of that trifecta that I was talking about.
Now to what you get in place of your d20 basics
Aspects. Remember where we talked about there really aren't attributes? That might be mind blowing. But look at it this way. If you're talking about a strength 20 fighter, another way to describe him might be A Hardy Fighter with Thick Thews. That's an aspect. And in cases where strength comes into play, that aspect gives you a leg up. (More on that later) They can also be used against you. The best aspects let you do both, which is why that example above about the thews wouldn't necessarily be a good aspect. This is the third part of the trifecta.
Levels, Advancement, and Playing the Game
The trifecta that I noted is powered by an economy of Fate points. You gain them by actually letting your aspect being used in a bad way against you. You use them to make use of your aspects for your good. Some stunts are also powered by them.
Of course there is the obvious in dice... instead of your polyhedral array, you use fate dice- that is d6s with a + on two sides, a - on two sides, and two blanks. You roll four of these, and get a number from -4 to +4. To this you add your skill, any modifiers from stunts, and a bonus of +2 for each aspect you use (invoke). You can also invoke your aspects to re-roll the dice.
You start out with a pool of Fate points equal to your refresh; your refresh also pretty much determines how powerful you are, as that's what you purchase your stunts from also!
When making an action, there are 4 basic choices - you can Overcome (achieve a goal with a skill), Attack (pretty self-explanatory), Defend (also self-explanatory), or Create an advantage. Creating an advantage means that you create an aspect that you can tag later in order to make yourself more likely to succeed. The thief might assess the lock to determine the type of lock, so that it's easier to pick. The fighter might assess his opponents style and determine a weakness he can exploit.
Combat is also a bit different in that your margin of success matters in your damage, and you have stress tracks (which are marked off according to the damage taken) and consequences (which are temporary aspects that reflect the wounds taken, and reduce the stress taken- they can be used as with any other aspect, except because the slant on it is so negative, it’s far more likely to be used to your character’s detriment.)
Summary
In the end, I think the biggest thing to get over is the change in paradigm. The fact that that game becomes more declarative, i.e. you say what you do, then use your character and surroundings to model it, than rules based, i.e. you look at your character to see your options, then describe the effect. It's really freeing once you get past that, and can model a wide variety of things without needed specific effects or rules for them.
