There comes a point where you'll want to start maxing your silver boxes, but having access to some of the emojis in the Series and Gold boxes will give you greater access to events and Daily Challenges.
There also gets to be a point where missions will ask you to use emojis from certain Disney properties (e.g. Frozen, Finding Nemo or Dory, etc.), and so working to get one from each of the properties proves to be quite helpful.