Instagram Direct didn't kill Snapchat. In fact, it solidified Snapchat as the preferred private sharing network for spontaneous sharing.
Instagram has established itself as a platform for an edited life. Sending a friend a bunch of quick, shitty pictures taints the main account. It also lingers.
The fundamental tenet of Snapchat is that your content disappears. It's as simple as that. Snapchat's business challenge is in developing ways to further innovate that private, ephemeral experience.
Instagram is for the curated life. Snapchat is for the imperfect life. It's as simple as that.