The panzerfaust was not a rocket weapon. The projectile was propelled by a black-powder charge in the tube itself - the propellant was not carried with the projectile, as it would with a rocket. Effectively, the panzerfaust is a mortar. The Panzerfaust suffered from a much more curved trajectory than the PIAT - just look at the length of the panzerfaust’s sights if you don’t believe me.
As has been said above, the PIAT suffers very much from traditional Army (‘ours is crap, while theirs is brilliant!’) syndrome. Yes, it was (reltively) difficult to cock, but it is almost invariably forgotten that it was designed as a blowback-operated, self-cocking weapon (although that sometimes didn’t work, according to one old PIAT gunner of my acquaintance).
As has been said, it had the advantages of no back-blast and a very low firing signature - Louis Hagen, in ‘Arnhem Lift’ describes how he repeatedly fired one from the attic of a house at SP guns passing beneath, without being detected (however, he didn’t hit very much due to his lack of training!).
This also brings us to another myth - it is often said that the PIAT could not be fired at a downward angle, due to the bomb falling out. This is completely untrue - a retaining clip held the bomb in place until fired (as proved by Hagen’s account above).
The warhead was sufficient to destroy even Tiger tanks and there are numerous examples of them doing so. It didn’t have as much punch as a Panzerfaust, but it had more oomph than a Bazooka. In the Far East, the weapon also saw sterling service as a bunker-buster, being infinitely preferable to having to crawl up to the bunker’s aperture with a grenade or satchel charge. The infrequently-encountered Japanese tanks were also easy meat for the PIAT, as demonstrated by Thapa VC at Imphal.
However, early versions of the PIAT bomb did not work well when striking a glancing blow against the target and according to my acquaintance, the bombs tailfin would come flying straight back at you when the bomb detonated (this side-effect actually killed his best friend in Normandy).
HE and smoke rounds were also supplied for the PIAT, though evidence for their use is scant.