One point that might be confusing some people is that the 70V amplifier is not always putting out 70V any more than a consumer power amplifier is always delivering its rated power. The "70V" is simply the maximum voltage output. And, of course, if the Volume control is set to zero in either case, the output is zero Volts and any speaker that you connect will be safe and silent.

We go through similar considerations for power wiring, the major difference being that we do supply a constant voltage. In power distribution we usually specify the circuit current because this is easily related to the fire risk. We could note that a particular Romex wire is good for 2400W, but this is true only if we also specify the circuit voltage. Or, we could simply give the minimum resistance that the circuit can support, spec the resistance of the light bulbs, heaters, appliances, computers, etc. connected to the circuit -- then calculate the equivalent resistance of the load. How much fun would that be?


In all cases the goal is to manage current and prevent damage. The specification and math scheme used is mostly traditional, given the context. In home audio four or eight Ohms is easy and traditional. In Pro audio 70V is much easier to work with, and current or watts is easy in home power wiring because of the constant voltage.
 
Edit: As a homework assignment calculate the equivalent resistance for a 50W load at 70V. This is the loading that our 70V amplifier expects to work into. This is why the 70V amplifier might freak out when an 8 Ohm load is connected.

Now consider the converse -- a 5W 70 V speaker connected to a consumer amplifier. Calculate the maximum output voltage of the consumer amp when driving an 8 Ohm load. There's no freak out, but the sound pressure level is a bit low from our 70V speaker. However, this is not necessarily the end of the world because a reasonably efficient speaker driven at a few hundred milliwatts, is "loud" for many livingroom listeners. (But, far from "thumping" for the disco crowd.)