Approaches vs Skills
Fate has three contexts in which you gain advantage or disadvantage (though Fate mostly gives out only advantage, and disadvantage by giving advantage to the obstacle/opponent). The three areas of Fate advantage are:
Randomness, which is almost always the die roll, in most Fate you cannot control that.
You, which is the things that define you, things you are always good at. Specific skills typically.
The situation, which defines how you do things in the context you are in.
The standard Fate example is 4dF+Skill+Stunts+Aspects. The standard FAE example is 4dF+Approach+Stunts+Aspects.
On the surface these are the same, but this breaks down when you use my (hopefully not contrived) groupings above. In the standard Fate Core game you are a skilled individual, and it doesn't matter how you get into a Fight, or why, if you have a +4 fight, you're going to be good in it. And if you jump out of an airplane and need to beat up a goon to steal their jetpack before you crash, being a better fighter is going to help. Whereas, if you have the stunt "Quick Jab - the first time you are attacked in close quarters in a round you may spend a Fate point to deal your defense successes as stress to your attacker", it's going to fair better than the person with the stunt "Hulk - whenever you are able to stand strong for more than one round, you gain +1 to attack for every person that attacks you this round". These both are pretty powerful stunts, but they are relevant based on the situation (falling from an airplane), not what you are doing (fighting).
In the standard FAE game skill is very different. If a strong fighter who can forcefully knock down walls and has the stunt "Great Club - when wielding a weapon at least as tall as yourself, you can get a +2 to attack with Forceful" is going to be a lot worse off than the small little rogue with stunts mostly like "Cat Burglar - You may use Quick instead of Sneaky when breaking into a building from the second story" might be far better off in the fight, since they may not be thought of as a fighter, but their high quick will pay off in this situation. The fighter might have the aspect "Bruiser" however, which will define that he's good in a punchup though, and might be enough for them to establish their status as a warrior with a Fate point.
So - in this role - 4dF define randomness, and then FAE's Stunts/Aspects or Fate Core's skills define You, and FAE's approaches and Fate Core's Stunts/Aspects make use of the situation. So - the main difference between FAE and Fate Core is what your character is built for. In Fate Core you build a character who is good at a list of specific things, and then their customization defines what situation they'll be good in. You might make someone with high Fight who will definitely be good in a fight, but they only have aspects for being in a noble court, and being a courtier, they might not be in their element on the battle field, but will always shine. Another character might have high Fight and a bunch of Aspects related to mercenary upbringing, and so their element is entirely during conflict, and they don't have a lot of area they can define.
In this way, I often feel like FAE is better for smaller groups, where different people can inhabit a fairly large set of situations each, and Fate Core is better for large groups, where each skill will be fairly narrow in their application, unless the GM focuses on one thing (usually combat).