If you want to use the Serre construction to produce a rank 2 bundle from a codimension 2 subvariety, the necessary condition is that the determinant of the normal bundle should be isomorphic to a restriction of a line bundle from the ambient variety. Equivalently, the canonical class of the subvariety should be restricted from the ambient variety.
For the abelian surface this is automatically fulfilled --- the canonical class is trivial hence restricted. But for generic surface in $P^4$ this is not true. For example, we can consider $P^2$ blown up in a point and its map given by the linear system $|2h-e|$. The image will be a cubic surface in $P^4$. But its canonical class $e-3h$ is not restricted since it is not a multiple of $2h-e$. By the way, this cubic surface is the simplest example of a residual intersection you are asking about (take just two quadrics $zx - vy =0$ and $ux - zy = 0$ in $P^4$ with coordinates $(x,y,z,u,v)$ and remove the palne $x = y = 0$).