The main differences are that the two-diode circuit requires a center tapped secondary, but has only one diode-drop in series with the voltage.
Silicon or Schottky diodes are cheap and small, so the number of diodes by itself is usually not the issue. Sometimes AC comes at you other than thru a transformer of your choosing. In that case, you don't get a nice center tapped secondary. Also note that the center tapped secondary is more costly since only half of it is used at a time. The four-diode circuit can get the same (minus one extra diode drop) full wave rectified voltage from just half of the same center tapped secondary. This usually outweighs the cost of the extra two diodes.
If you want a bipolar (both + and - supplies with common ground), then it takes a center tapped secondary and four diodes.