a^4 - b^4 = (a^2 + b^2) * (a^2 - b^2)

1 - insufficient. We don't know a^2+b^2

2 - insufficient. Again, we don't know a^2+b^2

1 and 2 together gives you

a^2 - b^2 = 16
(a+b)*(a-b) = 16
8*(a-b) = 16

a-b = 2, a+b = 8. We can solve for a and b and hence we can get
a^4 - b^4

So, C

a^4 - b^4 = (a^2-b^2) (a^2+b^2)

Statement I : We know one part, but we don't know (a^2+b^2)
hence insufficient

Statement II : (a-b)=8
Clearly insufficient

Combining I and II :

We know a^2 - b^2 = 16
or (a+b) (a-b) = 16
or (a+b) * 8 = 16
or a+b = 2

Hence we know a+b = 2
and a-b = 8

SO solve simultaneously to get values for a and b....