Impedance mismatch between speakers and amp

[Legion]

What effect does an impedance mismatch have on the actual sound coming out of speakers? If I hook up 4 ohm speakers to an amp expecting 8 ohms, other than overdriving the amp, what effect does it have on the sound?

[katzohki]

Difference in loudness. Maybe a bit of distortion depending on how hard your amplifier is working.

[TimFox]

Impedance mismatch, as that for coaxial cables and terminations at radio frequency or fast pulses, is not relevant normally to audio, where the signal wavelengths are much longer than the cables.
Power amplifiers (either vacuum-tube with an output transformer or solid-state with a direct connection) are normally designed to supply a constant voltage to the load, over a reasonable range of load impedance.  Solid-state amplifiers usually have an extremely low source impedance, much lower than the nominal load impedance.  The source impedance from a typical tube amplifier is higher, but still lower than the load impedance.
Lower impedance loads will draw more current than higher impedance loads.  The actual load impedance seen by vacuum tubes is somewhat critical to low distortion operation:  therefore, the output transformer usually has taps for different popular speaker impedances (4, 8, 16 ohms).  With solid-state, the most important question is the power dissipated by the output transistors at higher current than the original design.  A solid-state amplifier designed for a 4 ohm load will be safe with an 8 ohm load, but will deliver less power than into a 4 ohm load.

[Synthetase]

Another consideration would be power supply collapse. If the amp was designed to drive 8R loads, the supply may not be able to handle the additional current requirements of a 4R load and will collapse sooner, which will limit power output and could give you clipping (with the associated distortion products) much earlier, depending on how hard you're driving it.

[TimFox]

Yes, the output is limited by clipping, where the amplifier cannot supply more voltage to the load.  For a high-Z load, this is probably determined by the power supply voltage, but for a low-Z load the power supply maximum current may cause clipping at a lower voltage.  Also, amplifier protection circuitry may limit the current to protect the transistors.