Stories for everybody! Twitter is opening their Moments feature to the public, because Instagram Stories happened and now it's time to respond. The existing Moments tab will likely stay the same, but Moments created by individuals will be similar to moments currently created by brands and influencers. They will be accessible via a button on the user's profile that will take you to their Moments.
This strikes me as something they should have rolled out publicly from the beginning, but the Instagram launch may have accelerated their timeline.