Amplifier current vrs watts; why is current more important?

A speaker is an INDUCTIVE load.  What that means is that it proportionally resists the rise of voltage across it.  In Calculus, the voltage across an inductor is given as:

V = L x di / dt

L: inductance in henries and di / dt is the rate of change of current thru it.

When transients occur, the amp has to overcome the resistance of the speaker to increasing the voltage across it.  That is why it is difficult to drive inductive loads.  To be able to do that, amp has to be able to deliver loads of current.

Electronics 101: inductive loads.

PS: Music is not a straight sinusoid signal.  It is a MESS of transient signals.  A piano for example is almost like a steep 90 degree rise in signal.  To faithfully reproduce it, an amp must be able to supply LOTS of transient current, hence the need for an ENORMOUS power supply.

Having said that electrostatic speakers a different ball game.  They are capacitive loads.  Combine different frequencies and it is a miracle that modern amps actually work :-)

The issue is a little complex when discussing audio frequency modulated currents. Voice coil speakers use alternating electric current in copper coils in close proximity to the speaker magnets to create the mechanical motion of the speaker cone. Audio modulated AC current in any coil produces frequency dependent reactance and impedance not simple 8 ohm DC resistance that might be inferred by the speaker specification. The accuracy of overall sound reproduction is the ability of the amplifier to overcome the reactance and impedance across the audio spectrum otherwise sound distortion is introduced. There is also the issue of electrical resonance and the amplifier needs to be able to dampen such. In general all modern amplifier designs effectively deal with these issues unless you’re an electronics hobbyist building your own amp. 

But a monoblock amp like the "Legendary" old Mark Levinson ML2 monoblocks that are only 25w!! into 8ohms, will sound better into them, because it’s said they double their wattage for each halving of impedance all the way down to 1ohm with the current it can generate, so they will stay flat in frequency response, where the 500w’ers will sound like a tone control.

What is important is that the amp behave as a voltage source, not whether it can double full power as the load is cut in half. The latter is a subset of the former.

Of course, a good number of loudspeakers in high end audio don't even expect the amp to behave as a voltage source. That is why there are so many successful  zero feedback tube designs, which behave more like a power source.