I was expecting this to be about the empanada (although in hindsight that would be an odd title for an article about empanadas). It says here that the key difference is that an empanada's filling is cooked before being wrapped, whereas a paste's is not:

https://en.wikipedia.org/wiki/Paste_(pasty)#Differences_with...

And the pastry is different, but i suspect the pastry of both varies so much that wouldn't hold up empirically.