I was confused by this for awhile until I started hanging around a group of people who use SnapChat almost as much as texting. For the most part they share innocuous "here's what I'm doing" type photos. They use SnapChat because it's a really easy to send a photo to everyone on their list at the same time individually when they want to share a photo of what they're doing at that moment, with no feed or wall or comments to maintain later.

Thus, they really don't care if the photos are saved anywhere because they're not concerned with the privacy of their photos--at worst the snaps they share are unflattering. Their main concern is that they don't want to maintain any photos later, they just want to tell each other 'hi, here's what I'm doing' and forget about it.

In other words, to them SnapChat isn't a platform to share photos secretly, it's a platform to share photos momentarily. SnapChat even sells it as this:

>The images might be a little grainy, and you may not look your best, but that's the point. It's about the moment, a connection between friends in the present, and not just a pretty picture.

I get it now. I still don't like SnapChat at all (the UI is ugly, the UX is par, and I don't like the attitude of the company), but I see the user appeal and I see why the users SnapChat wants most will continue to use the app even though one of their friends may be keeping that picture they took of their thanksgiving turkey permanently.

How would that be an improvement over text messaging?

Thank god, finally some one explained it to me. I'm so released now.

Probably because SnapChat fixed that half a year ago.