I think the main difference between aspects and stunts is, that stunts represent WHAT you can do, while aspects explain WHY you do something. The latter is, from a story point of view, the more interesting.
In you examples above, the stunt "Brilliant Boxer" would give you a bonus any time you are boxing, plain and simple. If you are having a fight to free you sister, or even if your sister is just cheering you on, you canuse a fate point to invoke "I'll Always Be There for my Sister" to turn a fight around, if need be. You are activating every last bit of strength you have left, only because your sister is involved. But this would not only work on boxing, of course.
On the other hand, you are a "Brilliant Boxer", nothing is going to change that. But if your sister is held hostage and the bad guys tell you to throw the fight, you are going to throw the fight (and receive a fate point for the compell), because although you are one of the best, you wouldn't dare win this fight out of fear for your sister.
Mechanically, the main difference is, that stunts do not cost fate to activate, but they can't be compelled to gain fate points either like aspects can.
And lastly, aspects are telling something about your character. In this case, "I'll Always Be There for my Sister" is telling me your character has a sister that is very important to him. No mention of parents could indicate, that she is the only family left. What happened? Great Aspect to cover the characters entire past and what he learned from that, what makes him tick.
If another PC has the matching aspect "I'll Always Be There for my Brother", this would essentially become a fate point perpetuum mobile.
You should give the other players, especially the GM, an idea of what your aspects mean, what they should mostly be compelled for and where you most likely will invoke them. That way, there is going to be a lot less fuss about them.