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 $x = y = 0$). 

EDIT. Let me explain that the canonical class of the residual surface is almost never restricted. Indeed, let $X$ be the blowup of $P^4$ in $P^2$, let $H$ be the hyperplane class on $P^4$ (and its pullback to $X$ as well) and $E$ the exceptional divisor. Assume that the hypersurfaces we consider have degrees $a$ and $b$ respectively. Then they define divisors in linear systems $aH - E$ and $bH - E$ respectively, and the residual surface $S$ is the projection of their intersection. Since $K_X = -5H + E$, by adjunction 
$$
K_S = (-5H + E) + (aH - E) + (bH - E) = (a + b - 5)H - E.
$$
If $a,b \ge 2$ and $S$ is smooth then this is not proportional to $H$ (so is not restricted from $P^4$). Indeed, for $a,b \ge 2$ the linear systems $aH - E$ and $bH - E$ are very ample, so by Lefschetz theorem the restriction map $Pic X \to Pic S$ is injective, hence $H$ and $E$ are linearly independent on $S$.