One use is broad competency. This becomes more obvious in Fate Accelerated, that doesn't have skills, as such.
I play in a Fate Accelerated Space: 1889 game. My character has the High Concept: "Former Master Sergeant in Her Majesty's Service." You only get a few stunts, so you can't have one for each thing highly competent military personnel with a ton of experience in the field can do. But with this Aspect, I can get a bonus on anything you'd expect him to be good at - shooting, firing artillery pieces, driving vehicles, survival skills, field tactics and so on.
Another use is access to narrative, which, in Fate, matters, as you have to invoke that narrative to interact with things. My Master Sergeant can't invoke any supernatural narrative, as he's a very mundane character, by design. A "Fire Sorcerer of the 5th Circle" could invoke supernatural explanations for what they do.
Another related thing that an Aspect can get you that a skill or stunt can't is fictional positioning. My character's Trouble aspect is, "'Former' Master Sergeant in Her Majesty's Service" - while he says he is retired, nobody believes him - he is always assumed to be on some mysterious special detached mission or something, which gets him treated kind of like James Bond or something, for good or ill..