1. Orbital Refineries and their Vasari version, Matter Processors, work by sending cargo ships to Metal and Crystal Extractors in the same gravity well or in adjacent gravity wells to collect resources and take them back to the Refinery/Processor and therefore boost the total production output. Extractors built on orbital asteroids can feed up to 3 refineries and extractors built on neutral asteroids can feed up to 4 refineries, AFAIK.

The Advent version is completely different, however. Their 'refinery' works by unlocking the Resource Focus ability on Trade Ports, and activating this ability. Once activated, Resource Focus suspends trade ship operations and instead simply gives a percentage boost to the output of Metal/Crystal Extractors within the same gravity well.

The TEC get Orbital Refineries at Tier 4, the Vasari get Matter Processors at Tier 3, and the Advent get Resource Focus at Tier 4.

2./3. The Vasari's Jikara Navigator scout can capture neutral extractors early on and quickly, while the other factions can only do so using more expensive colony ships. It's far simpler than having to build a military, attack a neutral planet, defeat the planetary forces, colonize the planet, and then buy orbital extractors. The scout's ability together with the combo extractor technology and the early-access Matter Processors gives the Vasari an advantage over the others in resource production.