Facebook is where your relatives are, and Instagram is where your entire peer group is. Snapchat is where you can share with who you want (send private snaps to groups, and post stories to the public) while still seeing everyone's content. All while being reasonably confident that your photos/videos aren't being saved (or at least notified when they are, assuming somebody doesnt have an external camera ready).
All of that, plus the ads just feel fair. They exist, appear at the "right" times, and can easily be skipped if content isn't interesting.
I used to think snapchat was a fad. But as its evolved, I've become increasingly confident in their ability to stay relevant.
Edit: I didn't awknowledge your "no like count" point - also spot on. Only visible metric is views to your story, and that isn't public. Pressure to post something "amazing" just doesn't exist like other social networks.