(Is this something a non-US novice-class amateur needs to know about?)
Many contests use them as a convenient measure of how many different "regions" you've managed to contact. Since using countries as a contest multiplier isn't necessarily very accurate (within a certain radius, US can only easily reach Canada or Mexico or ocean, within the same radius, a station in central Europe could easily have their first 20 contacts come from 20 different countries) these equal-sized regions allow for a somewhat more equitable contest scoring system.
They also go into awards - there is a Worked All Zones award for the ITU zones, so while getting DXCC (century club - contacts with 100 different countries) may be much easier from Germany than it is from Kansas, both stations will have similar difficulty getting the Worked All Zones award, because the zones are more uniform worldwide than countries are.
Both are a means of identifying where you are in the world. CQ Zones is managed by CQ Magazine, and the ITU Zones are managed by the ITU. You can see more information about how they are defined in the two questions you linked. Some contests use them as an identifier of where you are, the most notable being the ITU number used in the IARU competition in July, and the CQ Zones are used by various competitions, usually those sponsored by CQ Magazine.
Both of these are an entry in to LOTW, which is where I became familiar with them. Most logging programs will ask for your CQ/ITU zone number. I'm sure there's other places they crop up as well.