Here $d(u)=2$ and $d(v) = 1$ so $d(u) - d(v) = 2-1 =1 $ so option $B$ eliminated.
here $d(u) = d(v) = 1 $ so $d(u) - d(v) = 0 $ so option $C$ eliminated.
Here $d(u)=1$ and $d(v) = 2$ so $d(u) - d(v) = 1-2 =-1 $ so option $D$ eliminated.
If you look at the above graphs then you would realize that we are just either calculating the shortest path of $s$ from either vertex $v$ or $u$(whichever is the shortest) and then just adding $1$ since both $u$ and $v$ are adjacent to each other so shortest path would increase by just $1$ edge distance.
Hence the difference between $d(u) - d(v)$ can never be greater than $1$.
So Option $A$ is the correct answer.