I've never used Snapchat, but from what I understand Snapchat basically sends people pictures with a time limit for looking at them. At that point, the client has perfect information, and it's up to the client to stop displaying the picture and remove it from system RAM, swap, CPU cache, GPU memory, and so on with 100% reliability after that timer expires, or the whole app is pointless. Since that's impossible unless you control every aspect of the system and encase it in self-destructing epoxy, the system is mostly pointless. (In that sense, it's like DRM. If you can view it, you can copy it.)
My understanding is that this is mostly used for sexting, which begs for me to ask this question: why are you sending naked pictures of yourself to someone you don't trust with naked pictures of yourself? Maybe work on those human relationships rather than outsourcing trust to some random company? Get off my lawn. Wheeze.
However, defaults matter. Even if you're sending a private picture to someone you completely trust, if you do it by email or mms, they'll likely leave that picture lying around inadvertently for someone borrowing their phone to accidentally and embarrassingly stumble upon. With snapchat, the picture will be automatically deleted unless the recipient takes explicit measures to save it. THAT is the feature, not security.
It still has some use. I don't want to keep the photos of the lunch my sister just ate; she doesn't want to keep the image of the bread I just baked. SnapChat is handy for these kinds of things where people do not care at all about security.
Obviously, if people are sending nudes and hoping those images will never find their way onto the Internet it's a bit more worrying. I guess we should be persuading people to think carefully before sending nude images of themselves to other people. (And educating youth about the perils of nudes).
First of all, look at what is actually successful for DRM in video games. The only surefire way to make sure someone is actually valid is to force them to authenticate(and get data from your server) to play the game. To do this they'll need to provide credentials(like a CD key) and then they'll get the content[1].
Now look at Snapchat. At no point do we need to fake having valid credentials. We are coming in and presenting valid credentials to Snapchat, my login and password, and it verifies those are correct and begins a sessions.
If you want an analogy to game DRM you need to look toward things like hacked clients in MMOs.
tl;dr Games want to prevent people without valid credentials from playing, we have valid credentials already(our own accounts). [1] This is really simplified and you can find many games, like the new SimCity, that the server interaction is small and simply emulated in the crack.
Maybe the problem is that I don't use snapchat so I don't understand why it's such a problem for them.