When you multiply them out, you get -ab and +ab. If you add those, you get 0. That's the definition of additive inverses... the +and- of the same number added together to get 0.
Just like 6 and -6 or 35 and -35 are additive inverses.
Basically, when you FOIL this out you get: a^2 + ab - ab - b^2
Combine like terms in the middle to get:
a^2 + 0 - b^2, or just...
a^2 - b^2.
That's where the special case she gave is derived from.