There are actual apps you can download that allow you to save a photo or video, not just the screenshot. I'm actually pretty baffled that people still don't know about them. Renders snapchat useless.
The concept of Snapchat is useless - it's literally DRM. It will work well enough for some folks, but fundamentally it simply isn't technically possible. At least not without a "trusted computing" model with remote attestation and so on.
You can prevent the actual bitstream from being copied with "secure" hardware, and most such hardware includes systems that verify output devices. Of course nothing stops you from taking a picture or video of the screen, but that's not making a digital copy. (Although, with a high resolution scanner, you could read every pixel and reverse the encoding and possibly end up with the same bit-by-bit value.)