var tileset = viewer.scene.primitives.add( new Cesium.Cesium3DTileset({ url: ..., pointCloudShading: { attenuation: true, maximumAttenuation: 2 } }));
The difference between the preview and the Sandcastle is that the former is using point cloud shading to enable point attenuation and eye dome lighting. See this code example for how to turn on and configure these properties:
var tileset = viewer.scene.primitives.add(new Cesium.Cesium3DTileset({url: ...,pointCloudShading: {attenuation: true,maximumAttenuation: 2}}));
Here's a Sandcastle example:
Let me know if that gets you the desired result!