Going by the physics I've learnt, matter-waves are particles in motion.

Considering that, aren't all waves the same? And, are waves just a simplification of phenomenon so that it's easier to understand? Can't phenomenon like interference be explained with the particle theory (considering calling photons as particles to be still valid)?

If you want particles to diffract and interfere, how would you explain it?

Well, ellastic collisions, and about interference:

Imagine that you hold two pen torches switched on and cross the light rays. At the point of intersection, we see a brighter region. Interference, perhaps can be explained on a similar basis in the microscopic level. Take photons to have a definite boundary, when they come across other photons and interact, their properties change.