Running out of voltage is harmless – the amp clips, it sounds bad, you turn the volume down, and life goes on. (The ostensible purpose of level meters is to help avoid clipping but we all know they just look cool.)

Running out of current can damage an amplifier. It can overstress the output devices and damage them, maybe catastrophically.

Fancy amplifiers have circuitry to protect against this, either nicely by momentarily reducing volume, or crudely by cutting off current output. The latter causes the amplifier’s output to momentarily present a high impedance to the speaker. That sounds bad and might damage tweeters, if flyback current from a moving woofer ends up in the tweeter since it can’t return to the amp.

Bob Cordell has a chapter in his amplifier book where he talks about how you don’t really need the protection circuitry. If the amp’s output stage has enough current capacity it becomes impossible to overstress it with a speaker of a given impedance.

My rule of thumb based on Mr. Cordell’s math is: if an amplifier has one pair of output transistors per 50W rated power into 8 ohms, it can safely drive any 4-ohm-rated speaker (regardless of its demand phase curve!) and will never be overstressed. Note this assumes dynamic speakers, not electrostatics.

For example if an amplifier is rated 150wpc @ 8 ohms, and it has 3 pairs of output devices per channel, it’s safe to drive a 4 ohm speaker. But if it only has two pairs of outputs, you’re on thin ice. (And you could double that number of output devices if you want to be safe into nominally 2 ohm loads, or halve it if you only care about nominally 8 ohm loads.)