In the book " Notes on Crystalline Cohomology" by P. Berthelot and A. Ogus, they introduced the cencept of PD-defferential operators in a complicate way, i.e. using dividied power hull.
However if I am understanding right, the sheaf of PD-defferential operators $D$ is just the enveloping algebra of the standard Lie algebroid of tangent bundle, with usual Lie braket. In other word, it is just the sheaf of algebra generated by the structure sheaf $\mathcal{O}$ and the tangent bundle $T$, subject to the module and commutator relations $f\cdot\partial=f\partial $, $ \quad\partial\cdot f-f\cdot \partial =\partial(f)$, $\quad\partial\in T, f\in \mathcal{O} $, and the Lie algebroid relations $\partial'\cdot \partial''-\partial''\cdot\partial'=[\partial',\partial'']$, $\quad\partial',\partial''\in T$. And this has no difference in char $p$ or char $0$.
My question is why P. Berthelot and A. Ogus use complicated construction via divided power hull, and say that there are difference in char $p$ and char $0$?
Consider the $l$-adic étale cohomology (with values in some "good" sheaf) of a "good" scheme over a characteristic $p$ field, where $l$ is a prime $\not= p$. In this framework, you have the so-called six operations (Grothendieck's definition) which are six functors related to morphisms between "good" schemes : the inverse image functor, the direct image functor, the proper (or extraordinary) direct image functor, the proper (or extraordinary) inverse image functor, the internal tensor product functor and the internal Hom functor. These functor have nice interplay when morphism are "good".
Now, we would like to have the same when $l = p$, but its is not the case. $p$-adic étale cohomology is not well behaved at all. To overcome this, we would like to have a nice replacement of the $p$-adic étale cohomology, with the six operations and their nice properties. To look for it, we can turn ourseleves to the charecteristic $0$ to seek inspiration. On $\mathbf{C}$ with analytic manifolds of over a good scheme over a field of characteristic $0$, you have the theory of $\mathscr{D}$-modules and holonomic $\mathscr{D}$-modules and they give you a good category to work in and define six operations etc. (For analytic case, see works of Mebkhout and Kashiwara.)
Now, you can define $\mathscr{D}$-modules in arbitrary characteristic (and it is of course done this way in EGA IV section 16) but in characteristic $p > 0$ you have an annoying phenomenon : take $X = \mathbf{A}_{\mathbf{F}_p}^1 := \textrm{Spec}(\mathbf{F}_p [T])$ the affine line over (the spectrum $S$ of) $\mathbf{F}_p$, and note $\mathscr{O}$ it's coordinate ring $\simeq \mathbf{F}_p [T]$. Consider the differential operator $D = \partial^p = \underbrace{\partial \circ \ldots \circ \partial}_{\textrm{$p$ times}}$, and let $f = T^{\alpha} \in \mathscr{O}$. If $\alpha < p$ we obviously have $D f = 0$. And if $\alpha \geq p$ we have $D f = \alpha (\alpha-1) \times \ldots \times (\alpha-p+1) f^{\alpha - p}$ and as $\alpha (\alpha-1) \times \ldots \times (\alpha-p+1)$ is the product of $p$ consecutive number, one of them is divisible by $p$, so that their product is zero in $\mathbf{F}_p$, showing that $D f = 0$. By linearity, we get that $D = 0$. This is annoying for various reasons, for instance because then an integrable connexion is not necessarily induced by a $\mathscr{D}$-module structure (see Grothendieck example of the so-called Gauss-Manin connexion, related to his $p$-curvature conjecture etc.)
To overcome this, you are obliged to work with divided powers, which will allow you to have, instead of the $\partial^p$ of the previous example, an operator $\partial^{[p]}$ such that $\partial^{[p]} \not = 0$. In fact, you are also obliged to do many manyn other things. See for this Pierre Berthelot's "Introduction à la théorie des $\mathscr{D}$-modules arithmétiques" as well as recent works from Daniel Caro on over-holonomic $F-\mathscr{D}_{\mathscr{X},\mathbf{Q}}^{\dagger}$-modules.