Rivets make a permanent fastening; Chicago screws can be unscrewed for adjustment or replacement; snaps are closures that open and close repeatedly. Choose the fitting for the construction, then use the setter or dies made for its exact type and size. A rivet-and-burr setter is a different tool from a cap-rivet or snap setter.
Choosing the right rivet requires matching two dimensions: cap diameter (proportionate to the leather weight and aesthetically appropriate for the application) and post length (which must match the combined thickness of the layers being joined). Assemble the actual leather layers and test nearby post lengths on matching offcuts before setting the finished piece. Removing a set rivet can mark the leather.
The fundamental difference is reversibility. Rivets are permanent — once set, the post is flared and cannot be removed without cutting or damage. Chicago screws are threaded and can be undone with a screwdriver, making them suitable wherever the connection may need to be removed or replaced. Use rivets for permanent structural joins; use Chicago screws where reversibility matters. Post length must suit the leather thickness for both.