The Panzerschreck and Panzerfaust, while seemingly a similar solution, are actually different in many ways. The panzerschreck was developed from the Puppchen once the Germans had a "doh" moment after capturing a bazooka. Somewhat ironic that a world leader in rocketry did not think of this first. I have my doubts about the actual wide-spread use of Puppchen, BTW, and the Germans did not manufacture many rounds for it once the Panzerschreck came into production. There is a famous photo of US troops messing with a Puppchen and one is actually holding a Panzerschreck round and it would not actually work in a Puppchen.
The Panzerschreck is a very expensive weapon overall compared to the panzerfaust. In fact, it is even rebuildable (tube replacement). Quite the opposite of the disposable Panzerfaust. It did not need batteries having instead a magneto type device for generating a voltage. The rocket is more expensive than a panzerfaust. The rocket features a point detonating fuse with spit-back that can attack sloped/oblique targets with some assurance. The panzerfaust had a base detonating fuse and it could have issues unless striking a solid surface. The actual amount of explosive in a Panzerfaust is quite impressive. Non-penetrating/ineffective hits on AFV could actually concuss the crew anyway.
The panzerschreck is a dedicated weapon system in the hands of a trained crew. The Panzerfaust was supposed to be a distributed weapon system, similar to grenades in that most soldiers could use them. The Germans clearly needed a means for its infantry to stand up to armored attack. As a weapon system, both had issues with range and danger for the users. The later model Panzerfaust, used at close range (less than the rated range), had a trajectory that would almost assure a hit and therefore 'accuracy'. Especially against tall sherman tanks.
The Germans built the panzerfaust to be as cheap as possible. They substituted explosive for metal in that the Panzerfaust used a soft-iron shaped charge liner. I believe the panzerfaust Klein, which was popular, did have an improved zinc liner improvement. copper, of course, was out of the question.
The Panzerfaust was a dangerous weapon and it was not to be put inside a trench but lay across the top. The base fuse apparently could lead to a detonation from artillery. Dropping one also had dangers.
A good webpage with more info than you probably want...
http://www.bergflak.com/psindex.html