It all depends on what you mean by "different". There are plenty of things you could do.
If they have the same functional forms:
- compare amplitudes.
- compare phases.
- compare other parameters determining the wave form. E.g., if the waves are gaussians you could compare their variance. If they are sinusoidal, you could compare frequence.
Should they be different functionally from each other, then you can
- compare average amplitudes.
- compare their spectra.
- compare average variances.
As Bernhard pointed out, you could also analyze the correlation function between the waves (provided such correlations should exist, of course).