What is the differences between scheme-theoretical intersection and set-theoretical intersection?

Let me restrict the discussion to the affine case, from which the general case is easily extended.
Given a ring $A$, the corresponding affine scheme is its prime spectrum $X=Spec(A)$. The crucial point, sometimes not sufficiently emphasized, is that there is a perfect correspondence between the ideals $I\subset A$ and the closed subschemes $Y=V(I)\subset X$. The scheme structure of $V(I)$ is obtained by identifying that set with $Spec(A/I)$.
Now if you have another such closed subscheme $Z=V(J)$ corresponding to the ideal $J\subset A$, their intersection is by definition the subscheme $T=Y\cap Z=V(I+J)\subset X$, corresponding to the ideal $I+J\subset A$. Set-theoretically the intersection consists of the prime ideals $P$ containing both $I$ and $J$, but the intersection scheme contains much, much more information. Let me show you this concretely.

An example Consider $X=Spec( k[S,T]) \subset \mathbb A^2_k$ , the affine plane over some field $k$.
Consider $Y=V(T)\subset \mathbb A^2_k$, the $S$-axis, and the family of suschemes $Z_n=V(T-S^n)\quad (n\geq 1)$.
The scheme-theoretic intersection $Y\cap Z_n$ is $Y\cap Z_n=V(T,T-S^n)=V(T,S^n)$. For all values of $n$, you obtain a subscheme of the plane with just one point: the origin, corresponding to the maximal ideal ${\frak m}=(S,T)$.
However all the intersections are scheme-theoretically different in pairs: the ring of functions of the subscheme $Y\cap Z_n$ is $k[S,T]/(T,S^n)=k[S]/(S^n)$, a k-algebra of dimension $n$ having a nilpotent radical $(S)/(S^n)\subset k[S]/(S^n)$ of dimension $n-1$.
The growth of the nilpotent radical with $n$ reflects the geometrical fact that the curve $T-S^n=0$ is ever more tangent to the $S$-axis with growing $n$.

Your article It seems very technical and I have certainly not read it. By very superficially browsing through it, I had the impression that the scheme theoretic intersection of equation (4.3) corresponds to the above description.

If $C_1, C_2$ are closed subset of a quasi-projective variety $X$ (according to Hartshorne's first chapter's definition), the set-theoretical intersection of $C_1$ and $C_2$ is the closed subset $C_1 \cap C_2$.

If $C_1, C_2$ are closed subschemes of a scheme $X$, the scheme-theoretical intersection of $C_1$ and $C_2$ is the closed subscheme $Y = C_1 \times_X C_2$. The underlying topological space of $Y$ is $C_1 \cap C_2$, but in this case also the structure sheaf is important.