Так а в чем разница-то?

Нашел объяснение на каком-то англоязычном форуме, по-моему, достаточно понятно:

In summary, while both "πάλι" and "ξανά" mean "again" in English, "πάλι" implies a return to a previous state or situation, while "ξανά" implies a fresh or new beginning. The choice between the two depends on the specific context and the intended meaning.