What is the difference between the leather cup and the poly cup, other than the price for the B&G Sprayer?
Really it is just preference. The B&G sprayers use to only come with leather cups. Some "old school" pest control people prefer the older leather cup. They worked great, but would wear out quicker than the poly cups. They made the alternative poly cup to last longer than the leather cup. Both work the same.
Where does the Leather cup part NP-271 go on a B&G Sprayer?
The Leather Cup (part# NP-271) goes inside of the Pump Cylinder. You can see the exploded view diagram of the B&G sprayer by clicking on this link B&G Sprayer product label. Scroll down to page 2 and the part you are asking about is #13 (NP-277 or NP-271) on the diagram.