They're completely unrelated.

Public Alerts is basically an emergency broadcast system used by governments to push notifications to everybody in an area... which might be an earthquake, but can be anything the authorities want to broadcast. For example, it saw some use in my city just last week due to a series of major storms passing through... and it was used during the Covid pandemic to advise of short-notice changes in the lockdown rules.

In contrast, Earthquake Alerts is a system for monitoring and warning of earthquakes specifically.