The mouth always seems to be what gives away the game the most. However, I don’t think it’s accurate to say that it’s because the meshes they’re using treat mouths as projections. That may still be true to some extent in games, but in big-budget Hollywood CGI, faces are extensively modeled to replicate the real thing, cavities and all. I think the problem is really more a result of how they animate them. Facial animation at this level is done with high-resolution motion capture tracking of a real person’s performance, and that movement is then mapped onto the “bones” and connections in the target model which are responsible for manipulating the mesh’s polygons.
For characters that don’t cross a certain “human-ness” threshold, it works great (Maz was animated with this technique in The Force Awakens, and I don’t recall anyone complaining about her performance), but despite the considerable amount of work that’s gone into making sure that these animation rigs closely model the constraints and extents of real human muscle and bone connections, they’re still not perfect. As a result, the mouth and cheek area is just never quite expressive enough during a highly emotional outburst, or it squidges around over the skull just a little too much when it moves. Given those problems, I think you’re right that Tarkin works as well as he does because Cushing’s mouth movements were always so restrained.
That said, the technique has come a long way since its early use in Polar Express (so many dead-eyed children… shudder), and it’s even improved a lot over Tron: Legacy’s Young Jeff Bridges/CLU rig from 2010 (which was impressive but occasionally really obvious). I think it’s at a point now where, when used judiciously and playing to its strengths, it’s good enough to fool most people - especially when, as @nungesser noted, they aren’t expecting someone to be CG. My mom also had no idea that Tarkin was digital, and it wasn’t until I started intentionally looking at him with a critical eye that I started catching the little things that gave it away.