What is the main difference between a bridge rectifier and a full wave rectifier?

Both the bridge rectifier and the full-wave rectifier achieve the same thing. They rectify the AC input on both opposing phases so as to minimize ripple time and voltage.

The difference is that a bridge rectifier consists of four diodes arranged in a bridge, so the input needs to only be single phase AC, while a full wave rectifier consists of two diodes, but needs a split phase AC source, such as provided by a center tapped transformer winding.

Also, the bridge rectifier presents two junction drops in the output, because there are always two diodes in series, while the full-wave rectifier presents only one junction drop in the output, because there is only one. It is a trade-off.