The difference is less the CIWS and more the amount of displacement that they have to spend on capabilities and the generality of those capabilities.
Even LCSs, effectively overpowered corvettes, have CIWS. They don't have the radar which can deal with threats further out than the 10 miles or so that their SeaRAM can deal with.
Frigates have more room to fit larger radars, power plants to feed them, and missiles to fire with them. Smaller frigates usually have 8-16 VLS tubes with ESSM or SM-2 to deal with threats 30+ miles out. Bigger ones, like the planned Constellation class, will have 32.
Destroyers are where you get enough room to start having strike and long range defense capabilities. The radar is now powerful enough to track ballistic missiles coming in from space, and they have missiles that can intercept them like SM-3. They also have extra VLS to store strike capabilities, like Tomahawk missiles. US Arleigh Burke class Destroyers have 96 VLS cells.
Cruisers have an interesting role. In the US navy, we don't really need them. We have a slightly larger destroyer hull with extra room for a fleet command center that we call cruisers to make congress happy. In navies that don't have carriers, though, their role is to be the main strike platform for the fleet. They'll have mounts for the largest mobile missiles in their arsenal, like the Russian P-700 Granit, which NATO called "Shipwreck".
Carriers are the most important part of the US Navy. Their power projection and sea control capabilities are unmatched. Each one carries their own airborne radar squadron, as well as 44+ strike fighters to not only defend it but to strike back at their foes. The carrier allows the fleet to know everything that happens within 500 miles, and destroy anything that it doesn't want there.
So, why don't we want to just have lots of cruisers? A mix of survivability and flexibility. While offensive capability scales linearly with tonnage, survivability falls to the cube-square law and only increases as the cube root of tonnage. Carriers are a special exception to this, because their aircraft are such a force multiplier. However, they're extremely expensive so very few countries can afford more than one or two. The US, however, is required by congress to have at least 10. For flexibility, it's often nice to have more platforms with less capabilites in more places than one platform with more capabilities in one place.