Let $A$ and $B$ be $R$-algebras. A Hasse-Schmidt $m$-derivation $D : A \to B$ is a tuple $(D_0, D_1, \dots, D_m)$ of $R$-linear maps $A \to B$ satisfying the generalized Leibniz law,
$$ D_k(xy) = \sum_{p + q = k} D_p(x) D_q(y) $$
If $M$ is an $A$-module, an $R$-linear map $D : A \to M$ is an $n$-th-order differential operator (in the sense of EGA) if, for each $x \in A$ the $R$-linear map,
$$ a \mapsto D(xa) - x D(a) $$
is an $(n-1)$-th-order differential operator. Also, a zeroth order differential operator is by definition an $A$-module map $A \to M$.
Now it is easy to show that if $D$ is a Hasse-Schmidt $m$-derivation then each $D_k$ is a $k$-th-order differential operator. In characteristic zero, I believe these operators (ranging over all $D$) generate the entire space of $n$-th-order differential operators. Is this true in positive characteristic?
These operators arise when studying jet schemes. If $X$ is a $k$-scheme, one definition of $J_m(X)$ is the scheme representing the functor,
$$ T \mapsto \mathrm{Hom}(T \times \Delta^m, X) $$
where $\Delta^m = \mathrm{Spec}(k[t]/(t^{m+1}))$. If $X = \mathrm{Spec}(A)$ then this functor restricted to affine schemes $\mathrm{Spec}(B)$ is exactly the functor,
$$ B \mapsto \mathrm{Der}_{\text{HS}}^m(A, B) $$
of Hasse-Schmidt $m$-derivations over $k$ as these correspond to $k$-algebra maps $A \to B[t]/(t^{m+1})$.
Another reasonable definition of the jet scheme is the total space of the jet bundle, $J^m(\mathcal{O}_X)$ which is defined as $\mathcal{O}_{X \times X} / \mathcal{I}_\Delta^{m+1}$ where $\mathcal{I}_{\Delta}$ is the ideal of the diagonal. This bundle is universal for $m$-th order differential operators meaning, $$ \mathrm{Hom}_{\mathcal{O}}(J^m(\mathcal{O}_X), -) = \mathrm{Diff}^m(\mathcal{O}, -) $$
In the case $m = 1$ we see that $J_1(X)$ is the total space of the tangent bundle of $X$ and $J^1(\mathcal{O})$ is a trivial extension of $\Omega_X$ by $\mathcal{O}$ so its total space is just a trivial extension of $J_1(X)$.
What, in general, is the relationship between these two schemes? There are obvious maps from the fact that Hasse-Schmidt derivations are differential operators but I don't see what more can be said.
