Amplifiers do what their name implies - they amplify a signal, namely the input voltage.
A 650W/8Ω at full power has a voltage at the speaker terminals of 72.11Vrms and would probably have a gain of 60x. So if you put a test tone of 1.2Vrms (72/60) into the input of the amplifier, did not connect a speaker, and measured the voltage at the speaker terminals, you'd see 72Vrms, but almost zero power. The meter will be very high impedance, probably close to 1GΩ.
When you now connect an 8Ω load to the terminals, 650W will be applied to the load. The current in the load will be
I = V/R = 72/8 = 9A.
A solid state amplifier simply supplies the voltage to the output terminals, and the load determines how much current flows*. If the speaker's impedance dropped to 4Ω at some point, it would pull 18A from the amplifier.
Power is voltage x current so 72V x 9A = 648W. (rounded because I'm lazy and used 72V, not 72.11V).
For a 4Ω load 72V x 18A = 1296W.
Now the source, whatever it is has some way of controlling the signal level, in this case the voltage. So if the test tone were reduced to 0.1vrms, the output voltage at the speaker terminals would be 6V (0.1x60), the current in the 8Ω load 0.75A (6/8) and the power supplied to the load 4.5W. So as you can see, the only thing that controls the power into a given load with a given amp is the signal level, and that is determined by the source material and the volume control (or gain) setting.
* All amplifiers have a current limit somewhere, and most do not double their output into half the load as in this example.