API Event Handlers - (as the name suggests) react on a specific event, e.g. you save some data in database (Data section), and you want to perform additional logic before or after that event, or you want to check whether you need additional logic at all.
API Services – is a independent peace of code, which you save on our side and can call directly or indirectly (as an external call, or from the page logic, or even from Event Handleer)