@gregm +1
High current is shorthand for delivering more current on demand. The P=IE formula remains the same.
An imperfect analogy would be gas engines. A Ford F-350 will go 60 mph. A Ford Fiesta will also go 60 mph. When towing a trailer with a 2 ton load, the F-350 will deliver more power on demand than the Fiesta. When going uphill, there will be less strain on the bigger engine – while the smaller engine might crap out.
Some speaker designs have low impedance dips at certain frequencies which require more instantaneous current delivery. The goal is to reproduce the authentic sound of the musical instrument: such as a thwack of a snare drum, the gut punch of a kickdrum, the shimmering of cymbals or the blaaat of a trumpet.
Matching the speaker and the amp is important. This is where transient response is involved.