This is because clipping planes are relative to the root tile's transform. This is described on the documentation page:
So for example, if you set the the clipping plane at (0,0,0), with a normal facing in the positive X axis, it would clip half the tileset (wheras the same plane would clip half the globe). The easiest way to get a plane to clip a tileset and the globe in the same location is to add a modelMatrix to the ClippingPlaneCollection on the tileset to reverse this transformation to make its coordinate relative to the center of the earth:
tileset.clippingPlanes = new Cesium.ClippingPlaneCollection({
planes : clippingPlanes,
edgeColor: Cesium.Color.WHITE,
modelMatrix: Cesium.Matrix4.inverse(tileset.root.computedTransform, new Cesium.Matrix4())
});
You might also want to use viewer.camera.pickEllipsoid instead of pickPosition to get more accurate results when not using terrain.