Specific to my knowledge base, timing is critical to applications in inertial sensors (IMUs that guide missiles, AR/VR, smartphone apps, and so much more). To that end, here is interesting work that DARPA is doing in developing inertial sensors that can navigate without GPS for significant amounts of time. Critical to that is accurate chip level atomic clocks.

https://www.darpa.mil/program/micro-technology-for-positioni...

Other uses as other posters mentioned is in traceability of measurements. Improving the means of measurement of the primary standard has downstream impact to secondary standards in metrology labs which ultimately (through transfer standards) would result in better calibrations of production systems.

Finally, an interesting thought I have is that in the future this could be an interesting way to figure out your absolute position on Earth. At a conference one of the NIST Boulder folks mentioned that the frequency of clocks in the development pipeline was so precise/stable that it could pick up time dilation effects from moving the clock from a desk to the floor! An instrument like that combined with accurate gravimetery data could be used to map the earth, find concentrations of natural resources, or possibly act as a standalone positioning system.

It’s a chicken and egg problem though. I don’t know if we will know all the uses for it until we build it, so instead of saying “why build it?” I think it makes sense to go and just build it :)