High Current vs high power

Hi!

Often you hear/read comments that the current matter more than the power (example Nait) and one should look for high current more than high power etc.?

Can anyone explain that or debunk the myth (my limited physics knowledge tells me that power and current are interrelated for the same voltage and impedance)?

Also, which amplifiers (pre power or Integrated) have 'higher current' than their peers at same power ratings? Is there any specification that shows the current capability of the amp which one can read and compare? like power, THD etc.?

Thanks
K