Does the new version of the DxO ONE App have any new features that only work with the latest version of the camera?

No, both generations of the DxO ONE camera can have the same features, if they update to the latest version of the DxO ONE App and firmware. What differentiates them is merely the removable back door for the second generation.

There may, however, be a difference between the features of the DxO ONE for iOS and Android, because the two products are not yet at the same stage of development.