When you save, you're saving exactly what would be uploaded to Snapchat, which is downsized for bandwidth reasons. You'll especially notice this on videos.

I'm looking at some of my saved snapchat videos on iOS and I am almost certain from the quality that there's no downsizing done here. Maybe if you send it to your story and then download it from there it'll be downsized to what exists on their server.

Move it to a desktop and you'll notice. I saved a number of videos directly from the camera view and it's much lower resolution than the built-in camera, like 720p or lower, which I didn't (disappointingly) discover until I got home.