  Let $x:M\to S^{n+p}$ be an $n$-dimensional submanifold in an
$(n+p)$-dimensional unit sphere $S^{n+p}$, $x:M\to S^{n+p}$ is called a
Willmore submanifold to the following Willmore functional: $$
\int_M(S-nH^2)^{\frac{n}{2}}dv, $$ where
$S=\sum\limits_{\alpha,i,j}(h^\alpha_{ij})^2$ is the square of the length of
the second fundamental form, $H$ is the mean curvature of $M$. In [13], author
proved an integral inequality of Simon's type for $n$-dimensional compact
Willmore hypersurfaces in $S^{n+1}$ and gave a characterization of {\it
Willmore tori}. In this paper, we generalize this result to $n$-dimensional
compact Willmore submanifolds in $S^{n+p}$. In fact, we obtain an integral
inequality of Simon's type for compact Willmore submanifolds in $S^{n+p}$ and
give a characterization of {\it willmore tori} and {\it Veronese surface} by
use of integral inequality.
