Amplifier Average Power vs. Peak Power
In audio amplifiers, power supply capacitors are being periodically recharged by the sources of power (transformer and bridge rectifier, the old fashioned way; or by more modern, fancy switching supplies.) The capacitors "smooth" out the voltage rail peaks and dips and can supply the peak instantaneous power to the load. Their voltage will droop a little but the negative feedback in the amplifier will keep the output linear with respect to the input source. Meanwhile, the power supply feeds the capacitors a bit later and recharges them back up. So only the average power needs to be supported for audio.

On the other hand, if you were to use the amplifier as a DC amplifier, which is effectively just a voltage-controlled power supply able to supply both plus and minus voltages to a load, then you would need to be able to support the peak power dissipation.

For example, a \$5\:\text{W}\$ audio amplifier for an \$8\:\Omega\$ speaker might use \$\pm 12\:\text{V}\$ rails and have to support two quadrants, each with \$1.8\:\text{W}\$ average dissipation. So this is perhaps about \$8.6\:\text{W}\$ total.

But if this were a voltage-controlled DC power amplifier, then the worst case dissipation would be \$4.5\:\text{W}\$ in the load and another \$4.5\:\text{W}\$ in the quadrant being used. A total of \$9\:\text{W}\$, which is slightly more for the power supply and then also quadrants that have to be able to dissipate almost 3X as much.