To me, Snapchat is like the modern version of email. The main reasons are because instead of sending a message to someone in a chat interface (initially), you come up with an attachment beforehand, and select the people you want to send it to.
To be honest, I just really like the idea of sending pictures of mundane things. It can be really fun to showcase small things that you find amusing, or to record small rants and then send them to your friends.
Snapchat has obviously been trying to be seen less and less as an app for sending naked pictures of yourself. First, they allowed removing the timer from pictures, then saving things in the chat, and then allowing saving Snaps, then removing the immediate auto deletion (only removing after 24 hours).
Instagram has been trying to compete with that, but nobody ever uses it, except I hear to send naked pictures of yourself. I wouldn't be surprised if most people weren't even aware Instagram had a Snapchat clone.
Despite Snapchat being seen as childish, I still really enjoy it. I don't like how they have been removing all of the fun elements of the app and making it less quirky, but I still think it is one of the best mobile ways of communication.