Anyone have specific examples of how IEEE 1588 is most commonly used in industry?
Synchronizing time stamping of data acquisition over an Ethernet network is an example.

The individual nodes will each have a clock, but there will be an unknown offset and a slight drift between clocks. If you don't care within milliseconds about synchronization then it's not  necessary but many applications are not that sloppy and need microseconds or better sync to achieve control or compensation.

Of course there are proprietary or ad hoc methods that achieve the same ends, but standards help ensure interoperability and may allow COTS boxes to be combined with bespoke ones.