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.

In general silver makes the solder stronger and has a higher melting point. We use it in high temperature applications such as downhole. Copper in the alloy lowers the melting point and makes it somewhat easier to work, and has some chemical advantages when soldering to copper conductors. Neither alloy has sufficient resistivity to matter much as far as the actual solder joint is concerned.

Contrary to what some other answers suggest, the Sn-Ag-Cu ternary eutectic melting point (217 °C) is below the 22/78 Sn-Ag (wt.%) eutectic of 221 °C and the 59/41 Sn-Cu eutectic of 227 °C. Those 10 degrees may be important.