The problem basically comes down to the fact that Snapchat needs to be able to discriminate between clients that are legit, and clients that are not legit. In other words, at some point Snapchat has to trust that it's distributing the server key (or cookie or whatever it uses in the future to allow clients to say "hey, it's me, you can trust me") to a legit client and NOT to a badguy client. This is not possible. Consider this issue taken to the extreme: even if you built Snapchat directly into the OS, and then signed, you could still root the phone and spoof the relevant parts of the OS to make it seem like you're a legit client when really you're not. Note that even pinning the cert doesn't work. You can still get around it by sniffing around the relevant buffers.
Spoofing the client basically guarantees that the API can be tricked into providing services that it was designed to avoid providing. For example, you can save all Snaps upon receipt. This is a trivial abuse. There are much graver implications that lead to much graver privacy concerns, but I hope you'll understand when I don't publicly announce them. :) Though perhaps Chris will tell you about them if you ask nicely.