Given that,
$ f(x+y) = f(x) f(y),$ and $f(5) = 4 $
Now, $ f(10) = f(5+5) $
$ \Rightarrow f(10) = f(5) f(5) $
$ \Rightarrow f(10) = 4 \times 4 $
$ \Rightarrow \boxed{ f(10) = 16} $
And, $ f(15) = f(5+10) $
$ \Rightarrow f(15) = f(5) f(10) $
$ \Rightarrow f(15) = 4 \times 16 $
$ \Rightarrow \boxed{ f(15) = 64} $
We can write $ f(5) = f[15+(-10)] $
$ \Rightarrow f(5) = f(15) f(-10) $
$ \Rightarrow 4 = 64 f(-10) $
$ \Rightarrow f(-10) = \frac{4}{64}$
$ \Rightarrow \boxed{ f(-10) = \frac{1}{16}} $
Then, $ f(10) – f(-10) = 16 – \frac{1}{16} = \frac{256-1}{16} = \frac{255}{16} = 15. 9375 $
$ \therefore$ The value of $f(10) – f(-10) = 15 . 9375.$
Correct Answer$: \text{B}$