What are the advantages of a synchronous switching regulator?

Every voltage drop in a regulator is a source of inefficiency. Synchronous regulators replace the normal schottky diode with a MOSFET that is open when the primary MOSFET is closed and vice versa to reduce or eliminate even the small inefficiency caused by the diode. This is true in both buck and boost regulators.

Ignacio has covered the main point. A lesser point but still useful to know is that a synchronous buck regulator (for instance) will be more stable than a non-sync buck generally. This is because the output voltage equals the input voltage x duty cycle and, if you have very good switches (low on resistance) then all you have to do to get decent output regulation is modify the duty cycle as the input voltage changes i.e. no feedback.

For instance, if the input voltage is 10V and you need 5V then the duty cycle is 50%. If the input voltage rose to 15V then your duty cycle should fall to 33.33% to obtain the same 5V output.

This is called "feed-forward" i.e. if volts are X then let duty be Y and there are no inate instabilities invoked.

Clearly, the output switching transistors will be hard-pressed to be perfect under all loading conditions and some negative feedback is required but it is usually much gentler than a non-sync buck and, is less liable to instability problems with load. In fact, given certain knowledge about the output current taken by the load, you can almost estimate a slight compensation scheme to counter the finite on-resistances of the switching transistors. Doing this would pretty much get decent regulation over a massive load range without worrying about any negative feedback and instabilities.