I think they have some of the most kinds of photoreceptors in their cones, something like 12 to 16, compared to our 3 (or rarely, 4). However, we and dogs perceive color in a way described by the oppositional process theory. It's kinda like extrapolating other colors for which which we don't have photoreceptors that activate most at that frequency. For example, yellow light falls between red and green frequencies. So our brains register the brightness of the light and the extent to which the red and green photoreceptors are activated proportionally, and it infers the color yellow.

Mantis shrimp have a far lower encephalization quotient, so it's unlikely they have the resources to do this kind of processing. I think they just go with the most activated photoreceptor, which would explain why they need so many. It's still entirely possible that they can see more colors than us, though. Also, it's almost a certainty that their visible spectrum is different from our own. Many animals can perceive ultraviolet or infrared light that we cannot.

Afaik Mantis Shrimp have more cones because their brains can't combine colours like human brains - ours can process the ratios of different wavelengths into colours, while they need a separate cone to detect each colour. I believe their total range is smaller than humans, but extends down into the IR wavelengths.