Particle weapons are simple, you're propelling solid matter (particles) at the target. Think of particle weapons like shotguns, the barrel is firing thousands of tiny little bullets, probably made of a superdense material like uranium, at the target.
Photon weapons are basically laser weapons, they're focusing photons (light, in layman's terms) at the target in order to create heat. When photons (light) collide with matter, their energy is transferred into that matter. In the same exact way a laser beam (which is actually a photon beam) burns through something, so does a photon cannon.
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