What do ohms of a speaker do?
Almost all audio amplifiers today are approximately voltage sources -- they have relatively low output impedance.  If you change the speaker impedance, to first order, you change the power delivered, which goes as $$P = V^2/R$$.

This applies both to the power delivered for a given input signal and volume setting, and also to the maximum possible power output.

You can't get infinite power out by lowering the speaker impedance.  There is a minimum possible impedance that the amplifier can effectively drive.  This will be specified in the data sheet, and will generally also allow the maximum power output.

Beyond that, there isn't much difference -- a similarly designed 8 ohm vs. 4 ohm speaker will operate mostly the same, as long as you drive it with the right type of amplifier.

There are often multiple ways to connect a speaker to an amplifier.  For instance, many multi-channel amplifiers can be connected in "bridge tied load" mode, where two ampliifer channels drive opposite signals to the speaker + and - (rather than having - at ground).  Other amplifiers always work this way, and still others allow multiple outputs to be connected in parallel to drive more output power.  Each of these modes may have a different minimum impedance and power vs. impedance curve, so make sure to look at the data sheet for the exact topology you use.