Comparators do not measure case diameters, they only take the headspace measurement. They are useful for setting up your dies, but are not useful for doing a complete chamber check. Like some case gauges mentioned above, they will accept an oversized round, so you could check the headspace length to get an accurate measurement of a fired case.
Comparators will provide a measurement, while the gauge itself gives you a simple and quick pass/fail reading. With our gauge, and case gauges as well, calipers could be used to measure the protrusion of the cartridge above the face of the gauge, or above the steps in the face of the gauge. So you can use them to get a measurement with the assistance of calipers.
Just using calipers can get you very accurate measurements to compare against the cartridge specs. There are only two real downsides to this. The first is time. If you are going to check your loaded rounds with calipers, you are going to be spending a lot of time doing it. The second downside is that you are only looking at the single measurement, not how it is tied in with all of the other measurements. All of the measurements on the gauge are tied to the same centerline. A bullet seated at an angle could measure just fine with the calipers, but fail our gauge and not chamber. A concentricity or runout measuring device would give you a good indication and measurement of problems like that. Our Ammunition Gauge will indicate this issue, and much quicker too.
With that said, if you do not own calipers, I would suggest getting them before any other measuring tools. They are useful for many applications, not just reloading.