A strong argument is given above on the heat equation; let me be more specific. The heat equation, one of the most basic in PDE and mathematical physics, already known to Fourier, is $$ L=\frac{\partial }{\partial t}-\Delta_x,\quad t\in\mathbb R,\quad x\in\mathbb R^n, $$ has the fundamental solution $$E= H(t)(4\pi t)^{-n/2}e^{-\frac{\vert x\vert^2}{4t}}, $$ i.e. $LE=\delta(x)\otimes\delta(t)$ (here the Heaviside function $H$ is the indicatrix of $\mathbb R_+$). It is easy to see that the $C^\infty$ singular support of $E$ is reduced to $0_{\mathbb R^{1+n}}$ whereas the analytic singular support is the hyperplane $t=0$. Since the function $E$ is $C^\infty$ except at $x=0,t=0$, one can see that it is indeed a flat function at $t=0,x=x_0\not=0$, i.e. all derivatives vanish at such a point. It is thus impossible to understand one of the simplest PDE using only analytic functions.
A more refined -yet classical- fact is related to the notion of well-posedness as defined by Jacques Hadamard. Loosely speaking, a PDE problem is well-posed whenever the solution can be controlled by the data or the sources via suitable inequalities. A typical example of a well-posed problem: the Cauchy problem with respect to a spacelike hypersurface (e.g. $t=0$) for the wave equation. A typical example of an ill-posed problem: the Cauchy problem for the Laplace equation. Although the latter has uniqueness properties, the analytic solutions given for instance by the Cauchy-Kovalewski Theorem are extremely unstable: you have $$ \partial_x^2 u+\partial_y^2 u=0,\quad u=e^{\lambda(x+iy)}, u(0,y)=e^{i\lambda y}. $$ The Cauchy data at $x=0$ are bounded by 1, whatever is $\lambda >0$, whereas the solution increases exponentially with $x>0$: no control of $u$ by its Cauchy datum could be expected. However the solutions are analytic and uniquely determined by the Cauchy datum. The analytic method given by the CK theorem provides analytic solutions which are unstable. The CK theorem fails to deliver stable solutions in that case. No understanding of stability phenomena (a very interesting physical property) for PDE is possible within the class of analytic functions and one should use much larger classes of functional spaces in which inequalities of well-posedness could be proven.
I could have mentioned another effect, for instance for the Cauchy problem for the Laplace equation: take an analytic Cauchy datum $\phi_0$, then CK provides an analytic solution. Now, perturb $\phi_0$ by a smooth non-analytic function $w$ and take as a datum say $\phi_0+\epsilon w$. Then there is no solution to the Cauchy problem since the very existence of a (say continuous) solution is forcing the data to be analytic. It is not difficult to prove that by Fourier transformation: the analyticity will be forced by the fact that you have to compensate the exponential increase by some exponential decay of the data, triggering analyticity for this data.
