As BCnrd points out, gluing cotangent complexes is a nontrivial thing. You might still ask whether it is really necessary for Illusie to work in the generality of a ringed topos. Would using a ringed space suffice? For standard deformation problems (deformation of a morphism or deformation of a scheme) working on the underlying ringed space would be enough. For more "interesting" deformation problems, like deformation of a morphism $X \rightarrow Y$, where $X$, $Y$, and the morphism are all allowed to vary, one needs something more sophisticated. Illusie constructs a ringed topos that encodes all of these data and then applies the machinery for ringed topoi that he has already developed.
A counterexample from one of my other MO answers seems to work again. Let $k$ be an algebraically closed field. Let $R$ be the ring of functions $f: k^2 \to k$ such that there exists a polynomial $\overline{f} \in k[x,y]$ with
1. $f(x,y) = \overline{f}(x,y)$ for all but finitely many $(x,y) \in k^2$ and
2. $f(0,0) = \overline{f}(0,0)$.
$R$ is reduced: If $f(x,y) \neq 0$, then $f(x,y)^n \neq 0$ for all $n$. $\square$
Every element $f$ of $R$ is either a unit or a zero divisor:
Case 1: $f$ is nowhere zero. In this case, $\overline{f}$ must lie in $k^{\ast}$, as otherwise $\overline{f}$ vanishes at infinitely many points and $f=\overline{f}$ at all but finitely many of them. So $f^{-1}$ is equal to the nonzero constant $\overline{f}^{-1}$ at all but finitely many points, and $\overline{f}^{-1} \in R$. So, in this case, $f$ is a unit.
Case 2: $f(x_0,y_0)=0$. Without loss of generality, we may assume that $(x_0,y_0) \neq (0,0)$. This is because, if $f(0,0)=0$ then $\overline{f}(0,0)=0$, implying that $\overline{f}$ vanishes at infinitely many points and $f=\overline{f}$ at all but finitely many of them, so we can find some other $(x_0,y_0)$ at which $f$ also vanishes. Let $\delta(x,y)$ be $1$ if $(x,y) = (x_0, y_0)$ and $0$ otherwise. Then $f \delta=0$ and $\delta \neq 0$, showing that $f$ is a zero divisor. $\square$
The set of functions vanishing at $(0,0)$ is clearly a maximal ideal of $R$; which we will denote $(0,0)$. We claim that $R_{(0,0)} \cong k[x,y]_{(0,0)}$. Proof sketch: We claim that $f = \overline{f}$ in the localization. To see this, let $g$ vanish at the finitely many points where $f \neq \overline{f}$, but $g(0,0) \neq 0$. Then $fg=\overline{f} g$, and $g$ is invertible in the localization. This shows that $f s^{-1} = \overline{f} \overline{s}^{-1}$ for any $f$ and $s$. $\square$.
Clearly, $k[x,y]_{(0,0)}$ is not classical.
I think this construction can clearly be generalized to make classical rings which have any local ring of dimension $\geq 2$ as a localization.
Best Answer
By comparing the tables of contents, the two books seem to contain almost the same material, with similar organization, with perhaps the omission of the chapter on excellent rings from the first, but the second book is considerably more user friendly for learners. There are about the same number of pages but almost twice as many words per page. The first book was almost like a set of class lecture notes from Professor Matsumura's 1967 course at Brandeis. Compared to the second book, the first had few exercises, relatively few references, and a short index. Chapters often began with definitions instead of a summary of results. Numerous definitions and basic ring theoretic concepts were taken for granted that are defined and discussed in the second. E.g. the fact that a power series ring over a noetherian ring is also noetherian is stated in the first book and proved in the second. The freeness of any projective modules over a local ring is stated in book one, proved in the finite case, and proved in general in book two. Derived functors such as Ext and Tor are assumed in the first book, while there is an appendix reviewing them in the second. Possibly the second book benefited from the input of the translator Miles Reid, at least Matsumura says so, and the difference in ease of reading between the two books is noticeable. Some arguments in the second are changed and adapted from the well written book by Atiyah and Macdonald. More than one of Matsumura's former students from his course at Brandeis which gave rise to the first book, including me, themselves prefer the second one. Thus, while experts may prefer book one, for many people who are reading Hartshorne, and are also learning commutative algebra, I would suggest the second book may be preferable.
Edit: Note there are also two editions of the earlier book Commutative algebra, and apparently only the second edition (according to its preface) includes the appendix with Matsumura's theory of excellent rings.