Maybe you have not noticed (and maybe noticed) - these schemes are completely identical, the only difference is, one conductor is connected to ground, or not. Just different pictures, and the diodes are connected between themselves are equally.
The first scheme is typically used when a single supply when minus power scheme must be common wire.
The second can be used the same way, if the scheme did not use the symbol notation common wire, but this common wire drafted as a separate conductor. Still the second scheme can be used in circuits with bipolar power, then the transformer must have an average output that is connected to the common wire circuit, and two outputs of the diode bridge will leave positive and negative power supply circuit.