What is the difference between a circular and a cat's-eye (slit) pupil?
Circular pupils are always functionally superior to vertical pupils; a slit does not correctly focus light from all directions whereas a circular pupil does. If you observe cats when they're hunting at dawn and dusk*, they have big, circular pupils; it's only when they're in bright light that the pupil shrinks to a slit. So why have vertical pupils at all? Because, as you mention they're capable of letting a more controlled range of light into the eye.

Thus vertical slits let you be active in a wider range of light conditions at the cost of poorer vision in bright light; whilst circular pupils continue to function well in brighter light but at the cost of not allowing such a large range of control over the amount of light entering the eye.

It is then easy to see why you get both kinds of pupil: circular pupils are favoured by animals that are typically active in bright light and need good vision under these circumstances; vertical pupils are favoured by animals that are primarily active in low light but need some ability to see in bright light.

*- it's often stated that cats are nocturnal, this is untrue; cats are crepuscular - that is they are most active at dawn and dusk.