A default route is the route that will be used is there's no other route that match the destination in the router forwarding table
a full routing table is a table which contain all the routes the BGP neighbor is aware off
a partial table is a table filtered (with route map, community...) so that only some specific routes are exchanged.
If you are connected to the Internet through a single ISP, a default route is enough, since there's only one possible path, so there's no point in having 500 000 routes in memory that all points to the same next-hop.
If you are connected through several ISP with BGP and want to always use the "best" path, than a full routing table make sens. In this case the default route will (almost) never be used since the router knows every possible destination with a specific route.
However a full routing table take some memory and also more CPU power (and time) to perform the lookup for all possible destinations.
When a vast majority of your traffic will take on path and only some specific destination will take another path, a partial table is more efficient.