  A differential operator of weight $\lambda$ is the algebraic abstraction of
the difference quotient $d_\lambda(f)(x):=\big(f(x+\lambda)-f(x)\big)/\lambda$,
including both the derivation as $\lambda$ approaches to $0$ and the difference
operator when $\lambda=1$. Correspondingly, differential algebra of weight
$\lambda$ extends the well-established theories of differential algebra and
difference algebra. In this paper, we initiate the study of differential
operators with weights, in particular difference operators, on lattices. We
show that differential operators of weight $-1$ on a lattice coincide with
differential operators, while differential operators are special cases of
difference operators. Distributivity of a lattice is characterized by the
existence of certain difference operators. Furthermore, we characterize and
enumerate difference operators on finite chains and finite quasi-antichains.
