They’re different.
* Full-wave uses two rectifiers and only one half of the secondary at a time. Each active half-secondary switches back and forth 120 or 100 times a second.
* A bridge uses four rectifiers and the entire secondary all the time. The current through the full secondary switches direction 120 or 100 times a second.
That’s the most basic level. To a first approximation, full-wave transformers need twice the voltage for the full secondary, since only half the winding is used at a time.
In more depth, if there is a first cap following the rectifier(s), the rectifiers only conduct for very brief intervals (a few milliseconds) when the cap recharges. These brief, high-current pulses can shock-excite the half-secondary, causing ringing from the abrupt cutoff of the unused half-secondary. The shape of that shock is controlled by the on-off ramp of the diode, and if there are charge-storage effects from a solid-state diode.
The rest of the time, after the first cap is recharged, the diodes are switched off, and the amplifier is powered from the charge in the first capacitor. This cap is steadily discharged until the next pulse comes along and recharges the cap all over again. This charge-discharge cycle happens 120 or 100 times a second.
There's a tradeoff in sizing that first capacitor. If you double the size, the voltage sags half as much ... but the inrush current is also doubled, too. If you really overdo it, the inrush current will be so large it pulls the circuit breaker.
A "choke-fed" supply has the diodes directly feed a special inductor rated for very high voltages. The diodes remain "on" for most of the AC waveform. But ... the special inductor has to tolerate a significant voltage kickback when the diodes do cut off ... this in turn can create ringing and possible voltage breakdown of the windings in the choke.
All of these supplies create very large current pulses that radiate into the air and are are transmitted back down the power cord of the amplifier, which turns the cord into an antenna radiating a 120 or 100 Hz pulse train, with harmonics extending throughout the audio band,
A general rule-of-thumb in power supply design is to minimize the "loop area" of the most powerful transmitters, which are the loop created by the transformer secondary, the diode array, and the first filter element. Minimizing the wire length, tightly twisting these wires, and keeping them as far away from the input circuit as possible is highly desirable. This is why running AC power to a front-panel switch is undesirable from a noise perspective.
There’s a simple emulator called PSUD that lets you play with different power supply topologies and circuit values, and lets you scale the load to a real amplifier.