Is there any special difference between "das letzte Mal" and "das vorige Mal " when they both mean "the last time"?
They both mean the same: "the last time", but – like in English –
das letzte Mal
can also mean: the very last time of something. For example:
In Hamburg besuchte ich meine Urgroßmutter das letzte Mal vor ihrem Tod.
Duden lists both attributes, "vorige", and "letzte" as synonyms for something that recently happened.
However in modern German "letzte" is used far more often. The usage of "vorige" declines:
As there is no general difference in the meaning there may be variations in regional usage. "Vorige" may be considered slightly more distinguished.
My understanding is that "das Vorige Mal" means the PREVIOUS time. It could mean the "last" time in a sequence of earlier events.
Das Letzte Mal LITERALLY means the LAST time. If the "previous" time will actually be the "last" or "final" time, then the two are equivalent.
But suppose an event is supposed to happen five times, and the most recent one was the fourth. Then the FIFTH event would be das LETZTE mal, not Das "VORIGE" (or the fourth).
There is even a threat: The next time you do X will be the LAST time. (Meaning I'll kill, or incapacitate you if you do x.) Here, letzte seems more appropriate than vorige.