Amplifier current vrs watts; why is current more important?

It's not more important. Power, Voltage, Current and Impedance are all related.

P=I^2*Z = V^2/Z

So depending on the impedance of your speakers I or V could be the limiting factor.

The late Charles Hansen used to say that an amplifier is basically a power supply design in which the supply is being modulated by an external musical signal.  

Wattage measurement is a static number that has no "time" variable, but in music, timing is everything.  Current is also a static variable without "time".  For example, if you say I want 7Watts but it could mean it will take 2 secs or 10 secs to get 7 watts.  In an orchestra crescendo, you need 7 watts right away.  A good amp will supply that 7 watts almost instantaneously whereas an inferior amp will not, therefore reducing the impact of the music.

A better metric for measuring an amp transient response is called "slew rate", that is how fast it responses to a change in input.

For any given amp, the slew rate will respond proportionally to the load (impedance) of the speaker.  That is the higher the load (higher impedance), the faster the slew rate.  That is why speakers with low impedance, you need higher power amp so that the slew rate will be adequate otherwise the music won't have any dynamic.  

As an analogy using cars, watt is like a car speed, whereas slew rate is car acceleration.  

Let's say you have two car: a good old American muscle car and a Ferrari.  Both cars may be able to do 150mph, but the Ferrari will accelerate to 150mph much faster than the muscle car which is what most people want.

Andy, are you addressing the Q of the OP or changing the topic? You say that both wattage and current are both static variables. How have you advanced his understanding about his Q?