What's the advantage of lead-free solder containing silver?
There are far more than two types available. RoHS and lead-free solders including silver have the following advantages and disadvantages:

Pros:


Higher melting point, higher working temperature.
Stronger bond, less susceptible to mechanical fatigue, more reliable joint.
Improved resistance to fatigue from thermal cycles.
Addition of an impurity to tin (silver, copper) reduces chance and/or rate of tin-whisker formation. (Note that silver itself can whisker in humid, hydrogen-sulfide environments.)


Cons: 


Too much silver can form inter-metallics that cause grittiness and formation of pimples on the solder surface.
Higher melting point = higher fabrication process temperatures.
Higher temperatures mean rework can be more difficult.
Stronger bond = more reliable, but also more brittle, having a lower ductility and higher Young's Modulus.
More expensive due to silver content.


While it is true that silver is a better conductor than most other metals, the resistivity of a typical solder joint is so low that any small gain in conductivity would matter only for very high-current applications. What usually matters more is the mechanical properties and assembly/rework-ability.