  In this paper, by using the "twisting technique" we obtain a class of new
modules $A_b$ over the Witt algebras $\mathcal{W}_n$ from modules $A$ over the
Weyl algebras $\mathcal{K}_n$ (of Laurent polynomials) for any
$b\in\mathbb{C}$. We give the necessary and sufficient conditions for $A_b$ to
be irreducible, and determine the necessary and sufficient conditions for two
such irreducible $\mathcal{W}_n$-modules to be isomorphic. Since
$\sl_{n+1}(\mathbb{C})$ is a subalgebra of $\mathcal{W}_n$, all the above
irreducible $\mathcal{W}_n$-modules $A_b$ can be considered as
$\sl_{n+1}(\mathbb{C})$-modules. For a class of such
$\sl_{n+1}(\mathbb{C})$-modules, denoted by
$\Omega_{1-a}(\lambda_1,\lambda_2,\cdots,\lambda_n)$ where $a\in\mathbb{C},
\lambda_1,\lambda_2,\cdots,\lambda_n \in \mathbb{C}^*$, we determine the
necessary and sufficient conditions for these $\sl_{n+1}(\mathbb{C})$-modules
to be irreducible. If the $\sl_{n+1}(\mathbb{C})$-module
$\Omega_{1-a}(\lambda_1,\lambda_2,\cdots,\lambda_n)$ is reducible, we prove
that it has a unique nontrivial submodule $W_{1-a}(\lambda_1,
\lambda_2,...\lambda_n)$ and the quotient module is the finite dimensional
$\sl_{n+1}(\mathbb{C})$-module with highest weight $m\Lambda_n$ for some
non-negative integer $m\in \Z_+$. The necessary and sufficient conditions for
two $\mathfrak{sl}_{n+1}(\mathbb{C})$-modules
$\Omega_{1-a}(\lambda_1,\lambda_2,\cdots,\lambda_n)$ and $W_{1-a}(\lambda_1,
\lambda_2,...\lambda_n)$ to be isomorphic are also determined. The irreducible
$\mathfrak{sl}_{n+1}(\mathbb{C})$-modules $\Omega_{1-a}(\lambda_1,
\lambda_2,...\lambda_n)$ and $W_{1-a}(\lambda_1, \lambda_2,...\lambda_n)$ are
new.
