What are the advantages of pulsed laser
There are really a couple of different classes of laser that matter for this:


CW : Runs continuously, this is your 6W non pulsed diode. 
QCW : Quasi CW, usually a thermal limit to the pulse length, good for tens of microsecond to tens of ms pulses, this is your 15W peak, 6W average device.
Q Switched : This is where the difference really appears, 6W average, but made up of 6mJ pulses 10s of ns wide at 1000 pulses per second, power during a pulse is 100s of kW, and the mechanism is as much ablation as classic bulk heat transfer. 


For most things the first two are much of a muchness, with the Q switched job being the one that behaves differently. For example copper is a bear to cut with a CW laser because it is so horribly thermally conductive, A Q switched job, because the pulses are so short but with enormous power makes a far better job of it. 

The bear with Q Switched lasers (Apart from the very much increased eye hazard), is that the pulses tend to burn the coatings on your optical chain.  

Only some laser technology Q switches in a useful way, usually YAG and some fiber lasers.