  In this paper we show that locally there exists a Willmore deformation
between minimal surfaces in $S^{n+2}$ and minimal surfaces in $H^{n+2}$, i.e.,
there exists a smooth family of Willmore surfaces $\{y_t,t\in[0,1]\}$ such that
$(y_t)|_{t=0}$ is conformally equivalent to a minimal surface in $S^{n+2}$ and
$(y_t)|_{t=1}$ is conformally equivalent to a minimal surface in $H^{n+2}$. For
some cases the deformations are global. Consider the Willmore deformations of
the Veronese two-sphere and its generalizations in $S^4$, for any positive
number $W_0\in\mathbb R^+$, we construct complete minimal surfaces in $H^4$
with Willmore energy being equal to $W_0$. An example of complete minimal
M\"{o}bius strip in $H^4$ with Willmore energy
$\frac{6\sqrt{5}\pi}{5}\approx10.733\pi$ is also presented. We also show that
all isotropic minimal surfaces in $S^4$ admit Jacobi fields different from
Killing fields, i.e., they are not "isolated".
