Aspects are the great leveler of the field. They let characters choose who they want to be.
Fate Points are the currency of Fate, they are the center point. A Fate point alone is worth a +1. But invoking an Aspect, which costs a Fate point, is worth +2. So, right away, Aspects double your exchange rate. +2 alone is big enough to be significant in most Fate games but if you have more than one Aspect that can apply to a situation (for example "Strongest Man in the Neighborhood" and "Gotta keep face" would both apply to lifting something heavy while in public) then they can all be tagged for 1 Fate point each. Sometimes a player can get +4 or +6. That's a big committment in Fate points but it lets players turn the tide in their favor.
Additionally, Aspects can be tagged (using a Fate point) to gain narrative effects. Using the above example, "Strongest Man in the Neighborhood" could be tagged to establish that the character in question has a bar bell set, or a lifting belt, or something a little stronger like body oil. This makes aspects extremely useful at giving characters things they should have according to their backstory.
Stunts are specific abilities. In most Fate games they change the way at a skill works or give a skill a bonus in narrow circumstances. Stunts cannot be tagged or invoked and they cannot be compelled. They just exist to provide their bonus, whatever it might be. Stunts might have a cost in Fate points depending on the Fate setting you are using.
Depending on the game system you select invoking an personal Aspect will always cost a Fate point. However temporary Aspects (such as Consequences) that pop up during a scene can generally be tagged once for free. Aspects universally cost the same (1 Fate point) when invoked (note the difference between "invoking" and "tagging").