Solutions

We have that $f+g$ is differentiable at $0$ when $f(x)=|x|$ and $g(x)=-|x|$. But neither $f(x)$ nor $g(x)$ is differentiable at $0$.

Moreover if both $f-g$ and $f$ are differentiable at $a$, then $g$ is differentiable at $a$ if $f(a)\ne0$.