Linear vs. quadratic dispersion relation In wave mechanics the dispersion relation between frequency $\omega$ and wave number $k$ is linear: $$\omega_n=c k_n$$
But in quantum mechanics, based on Schrödinger's equation, one can show that we have a quadratic relation between the two :
$$\omega_n=\frac{\hbar}{2m}k_n^2$$


*

*What is the implication of this difference? 

*Does it say something on the wave nature of wave-functions in quantum mechanics compared to e.g. EM waves?


Just looking for intuitive physical interpretations here.
 A: The wave mechanics dispersion relation you cite is for EM waves propagating in free space.  In other media, the dispersion relation is not necessarily linear (it can be quadratic or have some more complex dependence).  So in this context, there's nothing special about quantum mechanics.
More generally, the dispersion relation tells us about the phase speed of the wave and the group velocity:
$$v_\text{phase} = \frac{\omega_n}{k_n}$$
and 
$$ v_\text{g} \equiv \frac{\partial \omega_n}{\partial k_n} $$
So for example, in contrast to EM waves in free space, the particular quantum dispersion relation you cite will have a group velocity that depends on the wavenumber.  The quantum mechanics interpretation of this is that the particle's momentum will depend on its wavenumber ($p = \hbar k$).
A: That the phase speed can have a dependence on the wavelength/frequency of the wave.  For instance, a whistler mode wave can have a cubic dispersion relation at low frequencies.  In this limit, the higher(smaller) frequencies(wavelengths) propagate faster than the converse.  It results in a sort of "spreading out" of the wave modes.  This if often seen upstream of collisionless magnetized shocks in space.
You can also graph $\omega$ vs. $\kappa$ and show that the slope of the line at any given ($\omega$, $\kappa$) point corresponds to the group velocity and the ratio $\omega$/$\kappa$ corresponds to the phase velocity.
For instance, see the image below, modified from an figure in Krauss-Varban and Omidi, [1991]:

The $\omega$ vs. $\kappa$ diagram has been Doppler-shifted into a shock rest frame for this specific example, but it should illustrate my point.  Where the frequency goes to zero on this plot corresponds to a phase-standing mode (i.e., zero phase velocity) in this frame of reference.  In another frame, the same mode would have a finite phase velocity.  Where the slope of the blue line goes to zero, the wave's group velocity is zero (or group standing).  
Does that help?
References
Krauss-Varban, D., and N. Omidi "Structure of medium Mach number quasi-parallel shocks: Upstream and downstream waves," Journal of Geophysical Research 11, pp. 17,715--17,731, 1991.
