If I remember correctly, this came out of a OneWeek project - Microsoft's company-wide weeklong hackathon. Very cool to see a final published version of this!
It's not really intended for things you'd want to persist forever. If you'd like to, you can easily save the image or video. The idea is sharing your current experience with others in a way that you cannot really on Facebook or Instagram.
What I like about Snapchat is that things disappear. If someone posted a video on Instagram or Facebook every time their kid did something, not only would followers get annoyed, but it would almost diminish the entertainment value because there are countless more easily accessible on your page. What makes this one special?
You hit the nail on the head with Snapchat. You get to see something someone wanted to share. Maybe you shoot them an "lol" over text or maybe you bring it up next time you see them. It's wholly unobtrusive and each moment has equal value.